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\title[Standard conjectures in model theory]{
standard conjectures in model theory, and categoricity of comparison isomorphisms.\\
A model theory perspective.
%Standard conjectures on categoricity 
%of \'etale cohomology in model theory}
%of comparison isomorphisms
%in model theory
}
\author[misha gavrilovich]{misha gavrilovich
%\thanks{Some ideas were formed and formulated with help of Martin Bays.  $\tt{mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org}$. \href{http://mishap.sdf.org/hcats.pdf}{\tt http://mishap.sdf.org/hcats.pdf}.}
}
%\author{unfinished notes by misha gavrilovich\thanks{This text would be much less readable 
%without help of V. Sosnilo; some ideas were formulated with help of Martin Bays.
%Help in proofreading much appreciated; this is an unfinished darft I am still editing. 
%}}



%\title{???Standart conjectures on categoricity of functors in model theory}
%\author{unfinished notes by misha gavrilovich\thanks{This text would be much less readable 
%without help of V. Sosnilo; some ideas were formulated with help of Martin Bays.
%Help in proofreading much appreciated; this is an unfinished darft I am still editing. 
%}}

\address{National Research University Higher School of Economics, Saint-Petersburg\\
Institute for Regional Economics Studies of the Russian Academy of Sciences (IRES RAS)                                                                       
38 Serpuhovskaya st., Saint-Petersburg }
\email{mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org}
\urladdr{http://mishap.sdf.org/hcats.pdf}
\thanks{Some ideas were formed and formulated with help of Martin Bays.  
$\tt{mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org}$. \href{http://mishap.sdf.org/hcats.pdf}{\tt http://mishap.sdf.org/hcats.pdf}
}
%\thanks{ Support from Basic Research Program of the National Research University Higher
%School of Economics is gratefully acknowledged. This study was partially supported by
%the grant 16-01-00124-a of Russian Foundation for Basic Research.} 

\begin{document}

\maketitle

\begin{abstract}
We formulate two conjectures about \'etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. 

One conjecture says that there is a unique $\mathbb Z$-form
of the \'etale cohomology of complex algebraic varieties, up to $Aut(\mathbb C)$-action
on the source category;
put differently, each comparison isomorphism
between Betti and \'etale cohomology comes from a choice of a topology on $\mathbb C$.

Another conjecture says that each functor to groupoids from the category of complex algebraic varieties
which is similar to the topological fundamental groupoid functor $\pi_1^{top}$, 
in fact factors through $\pi_1^{top}$, up to a field automorphism of the complex numbers
acting on the category of complex algebraic varieties. 

We also try to present some evidence towards these conjectures, 
and show that some special cases seem related to Grothendieck 
standard conjectures and conjectures about motivic Galois group.  
\end{abstract}


\setcounter{tocdepth}{5}
{\small
\tableofcontents
}


\section{Introduction}

We consider the following question as it would be understood by a model theorist 
\begin{enonce*}{Question} Is there a purely algebraic definition of the notion of 
singular (Betti) cohomology or the topological fundamental groupoid of a complex algebraic variety?
\end{enonce*}
and formulate precise conjectures proposing that the comparison isomorphism of \'etale cohomology/fundamental groupoid 
admits such a purely algebraic definition (characterisation). 

Note that an algebraic geometer might interpret %is likely to understand %interpret
 the question differently from a logician and in that interpretation, the answer is well-known to be negative. 


For a logician this is a question about the desirable property of {\em categoricity}: a theory (i.e.~a collection of 
properties expressed in a language) of a model is called {\em categorical} iff any two models of the theory 
(i.e.~any two structures satisfying the properties) are necessarily isomorphic. 
In the approach of logically perfect structures [MoralesZilber], [Zilber] constructs a number of categorical theories 
associated with complex analytic functions and particularly exponentiation [BaysKirby], or rather their conjectural properties related to the field structure, 
such as Schanuel-type conjectures on transcendence of values, Kummer theory and Galois $\ell$-adic representations
(seen as being about the field automorphism action on the sequences of form $e^z, e^{z/2}, e^{z/3},...$).
In algebraic geometry, this kind of conjectures  occurs in the conjectural theory of the motivic Galois group 
and the conjectural theory of pure motives enabled by Grothendieck Standard Conjectures, 
e.g.~Mumford-Tate and Grothendieck periods conjecture.
Hence ``standard conjectures in model theory'' in the title refer both to the Grothendieck Standard Conjectures
and to the Zilber's standard conjectures in the approach of logically perfect structures 
claiming categoricity of certain theories 
associated with complex analytic functions.




We now explain our motivation in two essentially independent ways. \S1.1 explains how a model theorist 
would interpret the question above; \S1.2 views these conjectures as continuation of work in model theory on 
the complex field with pseudoexponentiation [Zilber, Bays-Zilber, Bays-Kirby. Manin-Zilber]
and its main goal is to make the reader aware of the possibilities offered by methods of model theory.

\subsection{\label{sec:1.1}How to interpret the question.}
Let us now explain the difference between how an algebraic geometer and a model theorist might interpret the question.

Let $H_{\mathrm{top}}$ be a functor defined on the category $\Var$ of algebraic varieties (say, separated schemes of finite type) 
over the field $\C$ of complex numbers; we identify this category with a subcategory of the category of topological spaces. 
We shall be interested in the case when $H_{\mathrm{top}}$ is either the functor $\Hsing:\Var\lra Ab$ of singular cohomology
or the fundamental groupoid functor $\pi_1^{top}:\Var\lra Groupoids$. 

An algebraic geometer might reason as follows. 
A purely algebraic definition applies both to $H_{\mathrm{top}}$ and $H_{\mathrm{top}}\circ \sigma$
where $\sigma:\C\lra\C$ is a field automorphism. Hence, to answer the question in the negative,
it is enough to find a field automorphism $\sigma:\C\lra\C$ such that $H_{\mathrm{top}}$ and $H_{\mathrm{top}}\circ \sigma$ %:\Var\lra  Ab$ and $H_{\mathrm{top}}\circ \sigma:\Var\lra Ab$
differ. And indeed, [Serre, Exemple] constructs an example of a projective algebraic variety $X$ and a field automorphism $\sigma$
such that $X(\C)$ and $X^\sigma(\C)$ have non-isomorphic fundamental groups. 

A model theorist might reason as follows. A purely algebraic definition applies both to $H_{\mathrm{top}}$ and $H_{\mathrm{top}}\circ \sigma$
where $\sigma:\C\lra\C$ is a field automorphism. Hence, we should try to find purely algebraic 
description  (possibly involving extra structure) of $H_{\mathrm{top}}$  
which fits precisely functors of form $H_{\mathrm{top}}\circ \sigma, \sigma\in \Aut(\C)$ with the extra structure. 
%such that any functor satisfying these properties is of form $H_{\mathrm{top}}\circ \sigma$ for some $ \sigma\in \Aut(\C)$.
We say that {\em such a purely algebraic description determines $H_{\mathrm{top}}$ (with the extra structure) 
uniquely up to an automorphism of $\C$.}

For $H_{\mathrm{top}}=\Hsing$ the singular (Betti) cohomology theory, a model theorist might continue thinking as follows.
The singular (Betti) cohomology theory admits a comparison isomorphism to a cohomology theory defined purely
algebraically, say $\ell$-adic \'etale cohomology theory. This is an algebraic description in itself. 
However, note that it considers the $\ell$-adic \'etale cohomology theory and the comparison isomorphism 
as part of structure. 
Thus an appropriate conjecture (see~\S\ref{htop:uniq}) %is that %there is a unique comparison
%each comparison isomorphism to \'etale cohomogy theory
%, up to $\sigma\i\Aut(\C)$:
gives a purely algebraic description of the family of comparison isomorphisms
coming from a choice of topology on $\C$
$$
\Hsing(X(\C)_{top},\Z)\tensor \Z_l \xra{\ \sigma\ } \Het(X(\C),\Zl),\ \ \ \sigma\in\Aut(\C)$$

For $H_{\mathrm{top}}=\pi_1^{top}$ a model theorist might continue thinking as follows. 
The profinite completion of the topological fundamental groupoid functor is the \'etale fundamental groupoid 
% $$\widehat{\pi_1^{top}}(X(\C)_{top})=\pi_1^{et}(X(\C)))$$
defined algebraically. This is an algebraic property of the topological fundamental groupoid
on which we can base our purely algebraic description %of topological fundamental groupoid 
if we include the \'etale fundamental groupoid as part of structure.
Essentially, this describes subgroupoids of the \'etale fundamental groupoids. 
Category theory suggests to consider a related universality property (see~\S\ref{pitop:uniq}):
{\em up to $\Aut(\C)$ action on the category of complex algebraic varieties},
there is a universal functor among those
 whose profinite
completion embeds into the \'etale fundamental groupoid,
and it is the topological fundamental groupoid.
Some technicalities may be necessary to ignore non-residually finite fundamental groups.
%%
%%---------In our conjectures these purely algebraic properties and extra structure
%%are {\em a comparison isomorphism} between the topological and \'\'etale functor:
%%\begin{itemize}
%%\item the comparison isomorphism of 
%%singular cohomology and %admits a comparison isomorphism to 
%%$\ell$-adic \'\'etale cohomology
%%\item the profinite completion of the topological fundamental groupoid functor is \'\'etale fundamental groupoid
%%\end{itemize}
%%




\subsubsection{Structure of the paper.}
In the introduction we explain our motivation in two essentially independent ways. 
\S\ref{sec:1.1} explains how a model theorist would interpret the
question above; \S\ref{sec:1.2} views these conjectures as a continuation of work in model theory
on the complex field with pseudoexponentiation [Zilber, Bays-Zilber, Bays-Kirby], see
also [Manin-Zilber, Ch. X.6.11-6.19] and its main goal is to make the reader aware of
the possibilities offered by methods of model theory.
In \S\ref{s:2} we begin by formulating two conjectures on \'etale cohomology. Conjecture~\ref{conj:ZHet} %2.1
says there is a unique there is a unique $\Z$-form of the \'etale cohomology of complex
algebraic varieties, up to $\Aut(\C)$-action on the source category; put differently, each
comparison isomorphism between Betti and \'etale cohomology comes from a choice of
 topology on $\C$. Conjecture~\ref{conj:Zmot} %2.2 
states a similar property for the Weil cohomology
theory restricted to a category generated by a single pure motive. In \S\ref{sec:2.2} 
Proposition~\ref{propo:2.3} and Statement~\ref{state:2.4} give examples of similar properties which hold for a
subcategory generated by a single abelian variety.
\S\ref{serre} contains a concise exposition of several conjectures on the motivic Galois group
which appear related to our conjectures; it follows [Serre,1.3]. \S\ref{sec:2.3} contains some
speculations about the model theoretic point of view on these conjectures.
\S\ref{pitop:uniq} states analogous conjectures for the etale fundamental group. In \S\ref{sec:3.1} we define
the notion of a $\pi_1$-like functor and state two conjectures saying there is a universal
$\pi_1$-like functor up to a field automorphism. In \S\ref{sec:3.2} we state two conjectures which
we hope to be within reach of current methods. In \S\ref{results} we list several partial positive
results which are implicit in model theoretic literature. We end by \S\ref{sec:3.4} which mentions
mathematical facts which go into the proofs of these results.



\subsection{\label{sec:1.2}Pseudo-exponentiation, Schanuel conjecture and categoricity theorems in model theory.}
Complex topology allows to construct a number of objects with good algebraic properties %defined analytically, %>with help of complex topology?,
e.g.~a group homomorphism %$ex_{\tau}:K^*\lra K^+$
$\exp:\C^+\lra \C^*$, singular (Betti) cohomology theory  
and the topological fundamental groupoid of varieties of complex algebraic varieties. 
%over $K$ (complex algebraic varieties). 

A number of theorems and conjectures says that such an object constructed topologically or analytically is ``free'' or ``generic'', for lack of better term, in the sense that it  satisfies algebraic relations only, or mostly, for ``obvious'' reasons 
of algebraic nature. %, say only those algebraic relations it was  due to its definition.?

Sometimes such a conjecture is made precise by saying that a certain
automorphism group is as large as possible subject to some ``obvious
obstructions or relations'' imposed by functoriality and/or homotopy theory.
Such an automorphism group may involve values of functions
 or spaces defined analytically or topologically. 
% due to how the object is defined.  

A natural question to ask is whether these conjectures are ``consistent" in
the sense that there do exist such ``free" objects with the conjectured
properties, not necessarily of analytic or topological origin. 

Methods of model theory allow to build such objects by an elaborate transfinite induction. 
In what follows we shall sketch results of [Zilber, Bays-Kirby] which does this for
the complex exponential function and Schanuel conjecture. 

Let us now explain what we mean by showing how to view Kummer theory, Hodge
conjecture, conjectural theory of the motivic Galois group, and Schanuel
conjecture in this way, i.e.~as saying certain objects satisfies algebraic relations only, or mostly,
for ``obvious'' reasons of algebraic nature.


\subsubsection{Grothendieck Standard Conjectures.% on algebraic cycles.
}
In words of [Grothendieck], 
`[these conjectures] would form a basis of the so-called theory of ``pure motives''
which is a systematic theory of ``arithmetic properties'' of algebraic varieties,
as embodied in their group of classes of cycles for numerical equivalence.'
For the purposes of this paper, it is important that these conjectures imply that 
there are many automorphisms of 
the Betti cohomology theory of the complex algebraic varieties, 
and are needed to formulate the conjectural theory of the motivic Galois group
 $\Aut^\tensor({\Hsing}_\sigma)$, cf.~\S\ref{sec:gal_conj}. 


A logician would wish to interpret the words of Grothendieck as follows. View a cohomology theory 
as a language which extends the usual theory ACF of algebraically closed fields 
by being able to talk about classes of algebraic cycles up to numerical equivalence; 
Grothendieck says this language is also able to talk about arithmetical properties
of algebraic varieties. 
%in other words, this language encodes classes of cycles 
%up to numerical equivalence. The group of automorphisms of the cohomology theory,
%i.e.~a model of a structure in this language of classes of algebraic cycles, encodes the arithmetic properties 
%of algebraic varieties. 
In model theory it is natural to assume (or wish) that such a structure is homogeneous and perhaps also categorical, which is a much stronger property.
In this way, it is tempting to think of the motivic Galois group as the automorphism 
group of such a structure, and therefore quite large. 
However, it is unclear how to make this point of view precise. 


%i.e.~the automorphism group of a structure in this language is large, 
%as this is needed to be able to contain the information about the arithmetic properties.    

In this paper we do not need the precise statements of the Standard Conjectures; see [Grothendieck, Kleiman] 
for details.
% of the conjectures, and [Serre] for an exposition 
%of the conjecutral theory of the motivic Galois group.




 

\subsubsection{Kummer theory.}
%The exponential function turns a $\Q$-linear relation between
%${\alpha_1},...,{\alpha_n}$ turned into a multiplicative relation
%between $e^{\alpha_1/N},...,e^{\alpha_n/N}$, $N>0$. 
An ``obvious way'' to make 
 $e^{\alpha_1/N},...,e^{\alpha_n/N}$, $N>0$
satisfy a polynomial relation is 
%for an algebraic (polynomial) relation between
%$e^{\alpha_1/N},...,e^{\alpha_n/N}$ 
is to pick ${\alpha_1},...,{\alpha_n}$ 
such that they satisfy a $\Q$-linear relation over $2\pi i$,
% between ${\alpha_1},...,{\alpha_n}$ 
which is preserved by $\exp$, or 
such that that $e^{\alpha_1/M},...,e^{\alpha_n/M}$
satisfy a polynomial relation for some other $M$. 

Kummer theory tells you these are the only reasons 
for polynomial relations between these numbers. 
This is stated precisely in terms of automorphisms groups as follows:

For any $\Q$-linearly independent numbers ${\alpha_1},...,{\alpha_n}\in \C$
there is $N>0$ such that for any $m>0$ it holds
$$\Gal(\Q(e^{\frac{\alpha_1}{mN}},...,e^{\frac{\alpha_n}{mN}},e^{2\pi i\Q})/
\Q(e^{\frac{\alpha_1}N},...,e^{\frac{\alpha_n}N},e^{2\pi i\Q}))\approx (\Z/m\Z)^n$$

%%
%%
%%Kummer theory tells you
%%that, except for finitely many exceptions, 
%%these are the only polynomial relations between 
%%$e^{\alpha_1/N},...,e^{\alpha_n/N}$ as  $N>0$ grows large. 
%%
%%
%%
%%
%%The exponential function turns a $\Q$-linear relation between
%%${\alpha_1},...,{\alpha_n}$ turned into a multiplicative relation
%%between $e^{\alpha_1/N},...,e^{\alpha_n/N}$, $N>0$. 
%%Think of this as the ``obvious reason'' for an algebraic (polynomial) relation between
%%these numbers. %$e^{\alpha_1/N},...,e^{\alpha_n/N}$. 
%%Kummer theory tells you
%%that, except for finitely many exceptions, 
%%these are the only polynomial relations between 
%%$e^{\alpha_1/N},...,e^{\alpha_n/N}$ as  $N>0$ grows large. 
%%
%%
%%
%%
%%This is expressed more formally in terms of automorphism groups:
%%%Kummer theory says that 
%%$$\Gal(\Q(e^{\alpha_1/N},...,e^{\alpha_n/N},e^{2\pi i\Q})/
%%\Q(e^{\alpha_1},...,e^{\alpha_n},e^{2\pi i\Q}))$$ 
%%has bounded index in $(\Z/N\Z)^{\mathrm{lin.deg._\Q}(\alpha_1,...,\alpha_n)}$
%%as $N$ goes to infinity.
%%%%
%%%%
%%
%%Roughly, this says that, except for finitely many,
%%polynomial % algebraic 
%%relations between
%%$e^{\alpha_1/N},...,e^{\alpha_n/N}$, $N>0$, 
%%are due to an algebraic relation between 
%%%with coefficients in 
%%$e^{\alpha_1},...,e^{\alpha_n}, e^{2\pi i\Q}$
%%or are due to  
%%a $\Q$-linear relation between 
%%${\alpha_1},...,{\alpha_n}$ turned into a multiplicative relation
%%by the exponential function. 
%%We view these  
%%$\Q$-linear relations preserved as ``obvious reasons''  
%%for the algebraic relations between 
%%between
%%$e^{\alpha_1/N},...,e^{\alpha_n/N}$, $N>0$,
%%and from this point of view, Kummer theory says 
%%that all relations between 
%%between
%%$e^{\alpha_1/N},...,e^{\alpha_n/N}$, $N>0$,
%%are due to obvious reasons, except for finitely many.
%%%%

%%An ``obvious reason'' for such an algebraic relation between 
%%$e^{\alpha_1/N},...,e^{\alpha_n/N}$ 
%%is a $\Q$-linear relation between 
%%${\alpha_1},...,{\alpha_n}$ turned into a multiplicative relation
%%by the exponential function. Kummer theory says these are the only ones, 
%%up to finitely many, an $N$ goes to infinity. 
%%

\subsubsection{Hodge conjecture}
Consider the Hodge theory of a non-singular complex projective manifold $X(\C)$. 
By Chow theory we know that $X$ is in fact a complex algebraic variety
and an easy argument using harmonic forms shows that an algebraic subvariety $Z(\C)$ defines
an element of 
$H(X,\Q)\cap H^{(p,p)}(X,\C)$
where $H^{(p,p)}(X,\C)$ is a certain linear subspace of $H^{2p}(X,\C)$ defined analytically. 


A topological cycle in $X(\C)$ defines an element of $H(X,\Q)$ which may lie in $H^{(p,p)}(X,\C)$.
An ``obvious reason'' for this is that it comes from an algebraic subvariety, 
or a $\Q$-linear combination of such. Hodge conjecture tells you that this is the only reason why it could happen.

\subsubsection{\label{sec:gal_conj}$\ell$-adic Galois representations and motivic Galois group $\Aut^\tensor({\Hsing}_\sigma)$}
Remarks below are quite vague but we hope some readers might find them helpful.  In \S\ref{serre} 
we sketch several definitions and conjectures in the conjectural theory of motivic Galois group following [Serre].

We would like to think that these conjectures say that the singular (Betti) cohomology theory of complex algebraic varieties
is ``free'' in the sense
that it satisfies algebraic relations only, or mostly, for ``obvious'' reasons of algebraic
nature. The theory of the motivic Galois group assumes that there are many automorphisms of 
the singular cohomology theory of complex algebraic varieties, and they form a
pro-algebraic, in fact pro-reductive ([Serre, Conjecture~2.1?], group.  
Conjectures on $\ell$-adic Galois representations, e.g.~[Serre, Conjecture~3.2?,9.1?], 
describe the image of Galois action as being dense or open in a certain 
algebraic group defined by the cohomology classes which the Galois action has to preserve (or is conjectured to preserve). 

Let us very briefly sketch some details. 

The conjectural theory of the motivic Galois group [Serre], also cf.~\S\ref{serre},
assumes that the following is a well-defined algebraic group:  
$$G_E=\Aut^\tensor\left({\Hsing}_\sigma:\left<E\right>\lra\QVect\right)$$
Here $\sigma:k\lra \C$ is an embedding of a number field $k$ into the field of complex numbers,
$E$ is a pure motive in the conjectural category $\Mot_k$
of pure motives defined over $k$, and 
$\left<E\right>$ is the least Tannakian subcategory of $\Mot_k$ 
containing $E$, and ${\Hsing}_\sigma$ is the fibre functor on $\left<E\right>$
corresponding to the singular cohomology of complex algebraic varieties
and embedding $\sigma:k\lra \C$. This is well-defined if we assume certain conjectures, 
e.g.~Standard Conjectures and Hodge conjecture [Serre, Grothendieck, Kleiman].

[Serre, Conjecture 3.1?] says that $G_E$ is the subgroup of
$GL({\Hsing}_\sigma(E))$  
preserving the tensors corresponding to morphisms
 ${\boldsymbol 1}\lra E^{\tensor r} \tensor E^{\vee\tensor s}$, $r,s\geq 0$. 
Think of these tensors as ``obvious relations'' which have to be preserved.



[Serre, Conjecture 3.2? and Conjecture 9.1?] describe the 
image of $\ell$-adic Galois representations in $G_E(\Ql)$.

Both say it is dense or open in the group of $\ell$-adic points of 
a certain algebraic subgroup of $GL_N$; we think of this subgroup 
as capturing ``obvious obstructions or relations'' imposed by functoriality 
of $\Hsing$.





\subsubsection{Schanuel conjecture: questions}

%Consider the complex exponentiation $\exp:\C^*\lra \C^+$. 
Schanuel conjecture says that for 
$\Q$-linearly independent $x_1,...,x_n\in \C$,
the transcendence degree of $x_1,...,x_n,e^{x_1},...,e^{x_n}$ is at least $n$:
$$\mathrm{tr.deg.}_\Q(x_1,...,x_n,e^{x_1},...,e^{x_n})\geq \mathrm{lin.deg.}_{\Q}(x_1,...,x_n) \eqno{\mathrm{(SC)}}$$
The bound becomes sharp if we use surjectivity to pick $x_2=e^{x_1}$, ..., $x_{i+1}=e^{x_i},...,x_n=e^{x_{n-1}}$
and $e^{x_n}\in \Q$:
$$\mathrm{tr.deg.}_\Q(x_1,e^{x_1},e^{e^{x_1}},...,x_n,e^{x_1},e^{e^{x_1}},...,e^{x_n})\leq n$$



Here ``an algebraic relation'' is a polynomial relation between $x_1,...,x_n,e^{x_1},...,e^{x_n}$;
%an ``obvious reason'' for such an algebraic relation is either a $\Q$-linear relation
%turned into a multiplicative relation by the group homomorphism $\exp:\C^+\lra\C^*$
an obvious way to make these numbers satisfy such a relation 
%$x_1,...,x_n,e^{x_1},...,e^{x_n}$ satisfy a polynomial relation such a relation 
is  to pick $x_i$ such that either $x_i=a_1x_1+...+a_{i-1}x_{i-1}$ 
or $e^{x_i}=a_1x_1+...+a_{i-1}x_{i-1}$ or $x_i=e^{a_1x_1+...+a_{i-1}x_{i-1}}$ 
 where $a_1,...,a_i\in \Q$ are rational.
%....
%or %that $x_i=e^{x_j}$ for some $i,j$, or, more generally, 
%we use surjectivity to pick $x_{i+1}$ such that
%$e^{x_{i+1}}$ satisfies an algebraic relation with  $x_1,...,x_i,e^{x_1},...,e^{x_i}$. 


%????Let us now list a number of questions about Schanuel conjecture and complex exponentiation
%a logician might ask. We then present a number of theorems in model theory which provide some answers.


Is Schanuel conjecture ``consistent'' in the sense that there is a {\em pseudo-exponentiation}, 
i.e.~a group homomorphism $\ex:\C^+\lra \C^*$ satisfying conjectural properties of 
complex exponentiation, in particular Schanuel conjecture?
Does there exist such a ``free''  pseudo-exponentiation  $\ex:\C^+\lra \C^*$, 
e.g.~such that a system of  exponential-polynomial equations has a zero only iff it 
does not contradict Schanuel conjecture? Can we build such an algebraic ``free'' object
without recourse to topology? 

Does every such ``free'' object come from a
choice of topology on $\C$, i.e.~is 
the complex exponential $\exp:\C^+\lra\C^*$ up an automorphism of $\C$? 

Note that the last question  is the only one which mentions topology. 
It turns out this difference is crucial: model theory says nothing about this question
while giving fairly satisfactory positive answers to the previous ones. 



\subsubsection{Schanuel conjecture and pseudoexponentiation: answers} 
The following theorem of [Zilber]  provides a positive answer for $\exp:\C^+\lra\C^*$. 
 For a discussion of the theorem and surrounding model theory see [Manin-Zilber, 6.16];
for a proof, detailed statements and generalisations to other analytic functions
see [Bays-Kirby, Thm.~1.2,Thm.~1.6; Thm.~9.1; also Thm.~8.2; Thm.~9.3] and references therein.
\def\ex{\mathrm{ex}}
\def\Ker{\mathrm{Ker}\,}
\begin{enonce}{Theorem}[Zilber]\label{Zil:preudoexp}
 Let $K$ be an uncountable algebraically closed field of characteristic $0$.

Up to $\Aut(K)$, there is a unique surjective group homomorphism 
$\ex:K^+\lra K^*$ % such that $\Ker \ex$ is the infinite cyclic group
%and 
\begin{itemize}
%\item[(ELA)]               F is an algebraically closed field of characteristic zero, and
%exp is a surjective homomorphism from Ga (F ) to Gm (F ).
\item[(SK)] (Standard Kernel) $\Ker \ex$ is the infinite cyclic group generated by a transcendental element
\item[(SC)] (Schanuel Property) $\ex:K^+\lra K^*$ satisfies Schanuel conjecture
\item[(SEAC)](Strong exponential-algebraic closedness)   
 any system of $n$ independent exponential-polynomial equations in $n$
     variables that does not directly contradict Schanuel conjecture has a regular
     zero, but not more than countably many
\end{itemize}
\end{enonce}

Call this unique group homomorphism $\ex:K^+\lra K^*$ {\em pseudoexponentiation} defined on field $K$.

In somewhat more detail, this can also be expressed as follows.

Let $K$ and $K'$ be two uncountable algebraically closed fields of characteristic $0$, 
and let $\ex:K^+\lra K^*$ and $\ex':K'^+\lra K'^*$ be group homomorphisms satisfying the properties above.

Then if there is a bijection $\sigma_0:K\xra{\,\,\approx\,\,} K'$, then there is a bijection $\sigma:K\xra{\,\,\approx\,\,} K'$
preserving $+$, $\cdot$, and $\ex$, i.e.~such that for each $x,y\in K$ it holds
$$\sigma(x+y)=\sigma(x)+\sigma(y),\ \ \sigma(xy)=\sigma(x)\sigma(y),\ \ 
\sigma(\ex(x))=\ex'(\sigma(x))$$

\begin{enonce}{Conjecture}[Zilber] If $\card K=\card \C$, %2^{\aleph_0}$, then 
then $(K,+,\cdot,\ex)$ is isomorphic to $(\C,+,\cdot,\exp)$.
%                   The unique algebraically closed field with pseudoexponenti-
%ation of power the continuum is isomorphic to (C, +, ·, exp), the complex field
%with exponentiation.
\end{enonce}

Our conjectures are direct analogues of the Theorem and Conjecture above
stated in the language of functors. Instead of the complex exponentiation
we consider the comparison isomorphisms between topological and \'etale 
cohomology, resp. fundamental groupoid functor. 
We hope that model theoretic  methods used by [Zilber] may be of use 
in proving these conjectures.



%6.16 Pseudoexponentiation. In the particular case of the class H(ex ) des-
%cribed above this author has carried out the steps 6.11 and 6.12 of Hrushovski’s
%construction (with some modifications). The resulting class of structures called
%algebraically closed fields with pseudoexponentiation, ACFExp, has the
%following properties:
%(i) ACFExp is axiomatizable by an explicit list of (not first-order) formulas,
%    stating
%(a) the validity of Schanuel’s conjecture and
%(b) that any system of n independent exponential-polynomial equations in n
%     variables that does not directly contradict Schanuel’s conjecture has a regular
%     zero, but not more than countably many;
% (ii) ACFExp is categorical in uncountable powers κ, that is, for every such κ
%      there is a unique, up to isomorphism, algebraically closed field with pseu-
%      doexponentiation of cardinality κ;
%(iii) An algebraically closed field with pseudoexponentiation carries a homo-
%      geneous pregeometry, in particular, any bijection between two bases of
%      the pregeometry can be extended to an automorpism of the field with
%      pseudoexponentiation.
%     A consequence of the theorem is that Schanuel’s conjecture is consistent
%with the field-theoretic algebra. The categoricity statement (ii) and homogene-
%ity statement (iii) strengthen this further on: Not only is Schanuel’s conjecture
%consistent, but along with other axioms, it also makes the algebra of the struc-
%ture uniquely nice.

%%Let $K$ be a field (non-canonically) isomorphic to the field $\C$ of complex
%%numbers. A choice of a locally connected locally compact topology $\tau$ on $K$
%%(almost) determines an isomorphism $K\xra{\tau_*}\C$ and thereby allows to
%%construct a number of algebraic objects defined analytically>with help of
%%complex topology?,~e.g.~a group homomorphism $ex_{\tau}:K^*\lra K^+$
%%($\exp:\C^+\lra \C^*$), 
%%singular (Betti) cohomology theory and the topological fundamental groupoid of 
%%varieties over $K$ (complex algebraic varieties). 
%%

%%journal% THE SECTION below IS REMOVED in THE JOURNAL VERSION
\subsubsection{Pseudoexponentiation: automorphisms groups} It is known that certain automorphisms groups 
associated with pseudoexp are largest possible in the following sense. 
The purpose of this section is to give a quick summary for the reader who might want to follow up 
our references to literature in model theory. 

We need some preliminary definitions. We say that tuples $a$ and $b$ in $K$ have {\em
the same quantifier-free type}, write $\qftp(a)=\qftp(b)$, 
iff they satisfy the same exponential-polynomial equations, and, moreover, 
the same exponential-polynomial equations with coefficients from $a$, resp. $b$, 
have a solution; see  [Bays-Kirby, \S6, Def.~6.7] for details.
Note that for a finite tuple $a$ in $K$, 
there is a minimal $\Q$-linear vector subpace $A\supset a$ such that $A\leq_\delta K$
and this $A$ determines $\qftp(a)$ (see below for the definition of $\leq_\delta$).
%%it's just immediate from richness wrt strong extensions, right?
%%and back and forth
%%if A and B are ss and isomorphic, then anything which exists over A also exists over B
%%so any A' <= A has the same qftp in the sense of [BK] as the corresponding B' <= B, because qftp is just about what exists over it
%%i.e. is part of the existential (infinitary) type
%%



We quote from [Bays-Kirby, Def.~6.1, Proposition~6.5].

\def\qftp{{\mathrm{qftp}}}
\begin{enonce}{Fact} Let $K$ be a field with pseudoexponentiation as defined above.  
\begin{itemize}
\item[QM4.] (Uniqueness of the generic type) Suppose that $C, C ' \subset M$ are countable
     closed subsets, enumerated such that $\qftp(C) = \qftp(C ' )$. If $a \in M\setminus C$
     and $a' \in M\setminus C '$ then $\qftp(C, a) = \qftp(C ' , a' )$ (with respect to the same
     enumerations for $C$ and $C '$ ).
\item[QM5.]  ($\aleph_0$-homogeneity over closed sets and the empty set)
Let $C, C' \subset K$ be countable closed subsets or empty, enumerated such
that $\qftp(C) = \qftp(C' )$, and let $b, b'$ be finite tuples from $K$ such that
$qftp(C, b) = \qftp(C ' , b' )$, and let $a \in cl(C, b)$. Then there is $a' \in  K$ such
that $qftp(C, b, a) = \qftp(C ' , b' , a' )$.


\item[QM5a.] ($\aleph_0$-homogeneity over the empty set)
      If $a$ and $b$ are finite tuples from $K$ and $\qftp(a) = \qftp(b)$ then there is 
a field automorphism $\theta:K\lra K$ preserving $\ex:K^+\lra K^*$ % $\theta\in  Aut(K^{\ex} )$ 
such that $\theta(a) = b$.
\end{itemize}
\end{enonce}

Note that it is an open problem to construct a non-trivial automorphism of $(\C,+,\cdot,\exp)$
which is not the complex conjugation. An easy argument shows that any continuous or measurable 
field automorphism of $\C$ is in fact either trivial or the complex conjugation. 

Note that the field with the pseudoexponentiation is highly homogeneous and has a large group of automorphisms.


\def\base{\mathrm{base}}
\subsubsection{Remarks about the proof.} We adapt [Manin-Zilber, 6.11-6.16];
see also [Bays-Kirby] for a detailed exposition in a more general case using different terminology.
Pseudoexponentiation is constructed by an elaborate transfinite induction. 
We start with an algebraically closed field $K_\base\subset K$ and a partial 
 group homomorphism $\ex:K_\base^+\longdashto K_\base^*$ 
and try to extend %construct 
the field and 
the group homomorphism such that 
it is related to the field in as free a
way as possible. 

Informally the freeness condition is described as follows:
% This is made precise in the following way, 
%and its exact form is what enables our construction:
\begin{itemize}\item[(Hr)]
the number of independent explicit basic dependencies {\em added} to 
a subset $X\cup \ex(X)$ of $K$ by the new structure is at most 
%{the number of independent explicit dependencies (codimension) 
%in $X$ in the new structure 
%must not be greater than 
%at most 
the dimension of $X\cup \ex(X)$ in the old structure.
\end{itemize}

This is made precise in the following way. %In the construction of pseudoexponentiation, 

{\em The new structure} is the group homomorphism $\ex:K^+\lra K^*$; 
{\em explicit basic dependencies in $X\cup \ex(X)$ added by the new structures} 
are defined as 
as equations $\ex(x)=y$ where $x\in X$. For example, for $X=\{x\}$  where %$x$ is such that 
$\ex(\ex(x))=x$, we do not regard $\ex(\ex(x))=x$ as a explicit basic dependency in $X\cup\ex(X)=\{x,\ex(x)\}$.  

%Schanuel conjecture is related to (Hr) as follows.
%Take $X=\Qx_1+...+\Qx_n+\Q\ex(x_1)+...+\Q\ex(x_n)$ to be the $\Q$-linear span of $x_1,..,x_n,\ex(x_1),...,\ex(x_n)$ in $K$.
%Then  
%the {explicit dependencies in} $X$ {added by the new structure} are of form 
%$\ex(\alpha_1x_1+...+\alpha_nx_n)
%=\beta_1x_1+...+\beta_nx_n$ where $\alpha_1,...,\alpha_n,\beta_1,...,\beta_n\in \Q$;
%among them, 
{\em The number of independent basic explicit dependencies} is the $\Q$-linear dimension 
$\mathrm{lin.dim.}_{\Q}(x_1,...,x_n)$; {\em the dimension of  $X$ in the old structure} is its transcendence degree
which is equal to $\mathrm{tr.deg.}(x_1,...,x_n,\ex(x_1),...,\ex(x_n))$.  

With this interpretation, (Hr) becomes Schanuel conjecture (SC). 

Define {\em Hrushovski predimension} $\delta(X):=\mathrm{tr.deg.}(X\cup\ex(X))-\mathrm{lin.dim.}_{\Q}(X)$. 
Say a partial group homomorphism $\ex:K^+\longdashto K^*$ satisfies {\em Hrushovski inequality
with respect to Hrushovski predimension $\delta$} 
iff 
%delta
%Cossider an algebraically closed field $K$ %be an algebraically closed field of characteristic $0$ 
%equipped with a group homomorphism $\ex:K^+\lra K^*$  such that 
for any finite $X\subset K$ it holds $\delta(X)\geq 0$. 
An extension $(K,\ex_K)\subset (L,\ex_L)$ of fields equipped with partial group homomorphisms 
is {\em strong}, write $K\leq_\delta L$, iff
all dependencies between elements of $K$ occurring in $L$ can be detected
already in $K$, i.e.~for every finite $X \subset K$,
$$  \min\{\,\delta(Y) : Y \text{ finite, } X \subset Y \subset K\,\} 
= \min\{\,\delta(Y ) : Y \text{ finite, } X \subset Y \subset L\,\}$$


We then build a countable algebraically closed field $(K_{\aleph_0},\ex_{K_{\aleph_0}})$ by
taking larger and larger  strong extensions $K_\base\leq_\delta K_1 \leq_\delta K_2 \leq_\delta ... $
of finite degree. 
If we do this with enough care, we obtain a countable algebraically closed field $K_{\aleph_0}=\cup K_n$  
and a group homomorphism $\ex_{K_{\aleph_0}}: K_{\aleph_0}^+\lra K_{\aleph_0}^*$ defined everywhere
which satisfies (SC) and other conditions of Theorem~\ref{Zil:preudoexp}.
For details see [Bays-Kirby, \S5] where it is described in terms of taking Fraisse limit
along a category of strong extensions.

Building an uncountable model requires deep model theory; see 
[Bays-Kirby, \S6] and $[\mathrm{BH^2K^2 14}]$. Let us say a couple of words about this. 
In the inductive construction above, being countable is essential: if we start with an uncountable field, 
we can no longer hope to obtain an algebraically closed field after taking union of countably many extensions
of finite degree. 
Very roughly, it turns out that we can construct composites of countable linearly disjoint
algebraically closed fields this way, and this helps to build an uncountable field with pseudoexponentiation 
and prove it is unique in its cardinality up to isomorphism. 


\subsubsection{Generalisations and Speculations.} 
[Bays-Kirby] generalises the considerations above 
in a number of ways. In particular, they construct pseudo-exponential maps of simple abelian varieties, including
 pseudo-$\wp$-functions for elliptic curve. 
[Proposition 10.1, \S10, ibid.] relates the Schanuel property of these to the Andr\'e-Grothendieck
conjecture on the periods of 1-motives. They 
 suspect that for abelian varieties the predimension inequality $\delta(X)\geq 0$
also follows from the Andr\'e-Grothendieck periods conjecture, but there are
more complications because the Mumford-Tate group plays a role and so have
not been able to verify it. [\S9.2, ibid.] says it is possible to construct a pseudoexponentiation 
incorporating a counterexample to Schanuel conjecture, by suitably modifying the Hrushovski predimension
and thus the inductive assumption (Hr). [\S9.7, also Thm.~1.7, ibid.] considers differential equations. 

We intentionally leave the following speculation vague. 
\begin{enonce}{Speculation} Can one build a pseudo-singular, or pseudo-de Rham cohomology theory, or a pseudo-topological fundamental group functor 
 of complex algebraic varieties, or an algebra of pseudo-periods
which satisfies a number of conjectures 
such as the Standard Conjectures, the conjectural theory of the motivic Galois group,
the conjectures on the image of $\ell$-adic Galois representations,  Andr\'e-Grothendieck periods
conjecture, Mumford-Tate conjecture, etc.? 
\end{enonce}


%%journal% THE SECTION below IS REMOVED in THE JOURNAL VERSION
\subsection{A glossary of terminology in model theory.} 
%Our conjectures are direct analogues of categoricity conjectures and theorems in models theory.
% A categoricity theorem says that  any two models satisfying a certain property are isomorphic.
%a unique, up to isomorphism, model satisfying a certain property. 
%Let us now explain this in more detail. 

We give a very quick overview of basic terminology used in model theory, to help the reader
follow up references to model theoretic literature.  
See [Tent-Ziegler; Manin-Zilber] for an introduction into model theory.


In logic, a property is called {\em categorical} iff 
any two structures (models) satisfying the property are necessarily isomorphic. 
A {\em structure} or {\em a model} is usually understood as a set $X$  equipped
with names for certain distinguished subsets of its finite Cartesian powers $X^n$, $n>0$, called {\em predicates},
and also equipped with names for certain distinguished functions between its finite Cartesian powers. 
Names of predicates and functions form a {\em language}.
{\em First order formulas in language $L$} is a particular class of formulas which provide
names for subsets obtained from the
$L$-distinguished subsets by taking finitely many times intersection, union,
completion, and projection onto some of the coordinates; a formula $\varphi(x_1,..,x_n)$ 
defines the subset $\varphi(M^n)$ of $M^n$ 
consisting of tuples satisfying the formula. A {\em theory in language $L$} is a collection of formulas in language $L$.
{\em A model} of a theory $T$ in language $L$ is a structure in language $L$ such that 
for each $\varphi\in T$ $\varphi(M^n)=M^n$ where $n$ is the arity of $\varphi$.

{\em The first order theory of a structure} consists of all possible names (formulas)
for the subsets $M^n,n\geq 0$, i.e.~formulas $\varphi$ such that $\varphi(M^n)=M^n$. 


A {\em categoricity} theorem in model theory usually says that any two %(usually assumed uncountable) 
models of a first order theory of the same uncountable cardinality are necessarily isomorphic,
i.e.~if there is a bijection between (usually assumed uncountable) models $M_1$
and $M_2$ of the theory, then there is a bijection which preserves the distinguished subsets
and functions. A theory is {\em uncountably categorical} iff it has a unique model, up to isomorphism, 
of each uncountable cardinality. 

The {\em type $\mathrm{tp}(a_1,\!...,a_n)=\{\varphi(x_1,\!..,x_n):\varphi(a_1,\!...,a_n)\text{ holds in } M\}$ 
of a tuple $(a_1,..,a_n)\in M^n$} is the collection of all formulas satisfied by 
the tuple $(a_1,..,a_n)$. A {\em type in a theory} is the type of a tuple in a model of the theory. 
The {\em type $\mathrm{tp}(a_1,...,a_n)=\{\,\varphi(x_1,..,x_n):\varphi(a_1,...,a_n)\text{ holds in } M\}$ 
of a tuple $(a_1,..,a_n)\in M^n$ with parameters in subset $A\subset M$} is the collection of 
all formulas with parameters in $A$ satisfied by 
the tuple $(a_1,..,a_n)$. A {\em type in a theory} is the type of a tuple in a model of the theory. 
Informally, the type of a tuple is a syntactic notion playing the role of an orbit of $\Aut^L(M)$ on $M^n$, 
e.g.~in a situation when we do not yet know whether non-trivial automorphisms of $M$ exist,
for example because $M$ has not been completely constructed yet. 


In an uncountably categorical first order theory with finitely many predicates and functions
the number of types is at most countable, and the number of types with parameters in a subset $A$ 
has cardinality at most $\card A + \aleph_0=\max(\card A,\aleph_0)$. 


 

 
 

%Kummer theory says that the multiplicative group structure
%"sees" all the restrictions on the Galois action on the roots $a^{1/n},..,c^{1/n}$, up to finite index. 
%The conjectural description of the Galois action on the torsion points of an Abelian variety provided by 
%Mumford-Tate conjecture may be viewed as saying that
%the Hodge tensors on the cohomology forming the Mumford-Tate group "see"/capture all the restrictions on the image of 
%Galois action on the torsion points, up to finite index.
%
%From this point of view, our conjectures may be described as saying the fundamental groupoid ":sees"
%the restrictions on the Galois action on the preimages of a point of a variety $X$ in finite \'etale covers of $X$.
%







\section{\label{htop:uniq}Uniqueness property of comparison isomorphism of singular and \'etale cohomology of a complex algebraic variety}

%%>fixme?: what's below is kind of obvious, a verbose explanation of something obvious to some

\subsection{\label{s:2}Statement of the conjectures}
A {\em $\Z$-form} of a functor $H_l:\VV\lra\ZlVect$ is 
%a functor $H:\VV\lra\QVect$ equipped with an isomorphism 
a pair $(H,\tau)$ consisting of a functor  $H:\VV\lra\ZVect$ and an isomorphism 
$$
H\tensor_\Z \Zl \xra{\tau} H_l 
$$
of functors. 

An example of a $\Z$-form we are interested in is given by the comparison isomorphism between 
\'etale cohomology and Betti cohomology, see [SGA 4, XVI, 4.1], also
[Katz,p.23] for the definitions and exact statements. 

%Let $\VVC$ be the category of separated $\C$-schemes of finite type, and 
Let 
$\Het:Schemes\lra \ZVect$ be the functor of $\ell$-adic \'etale cohomology, and let $H_{sing}:Top\lra \ZVect$ be 
the functor of singular cohomology. For $X$ a separated $\C$-scheme of finite type there is a canonical {\em comparison isomorphism}  
$$H_{sing}(X(\C),\Z))\tensor \Zl \approx \Het(X,\Zl)
.$$
This defines a $\Z$-form of the functor of $\ell$-adic \'etale cohomology $\Het(-,\Zl)$ restricted to the category of separated $\C$-schemes of finite type. 

Let $K$ be an algebraically closed field, let $\VVK$ be a category of varieties over $K$. %, and 
%let $\Hl:\VVK\lra\ZlVect$ be the functor of $\ell$-adic cohomology. 
A field automorphism $\sigma:K\lra K$ 
acts  $X\mapsto X^\sigma$ on the category $\VVK$ by automorphisms. Moreover, for each variety $X$ defined over $K$,
 a field automorphism $\sigma$ defines an isomorphism $\sigma_X:X\lra X^\sigma$ of schemes (over $\Z$ or $\Z/p\Z$), and hence 
$$\Het(X,\Zl)\xra{\sigma_*}\Het(X^\sigma,\Zl)
.$$

This defines an action of $\Aut(K)$ on the %trivialised 
$\Z$-forms of $\Hl$:
$$(H,\tau)\longmapsto (H\circ\sigma, \tau\circ\sigma^{-1}_*)$$
$$ 
H(X^\sigma)\xra{\tau\circ\sigma^{-1}_*} \Het(X,\Zl)
.$$

%Recall [Kleiman] defines a Weil cohomology theory as a functor $H:\Var\lra\Qvect$ from varieties to vector spaces
%with certain extra structure which includes the cycle map, the Poincare pairing
%and Kunneth decomposition formula. Accordingly, we may define the notion of a $\Q$-form 
%of a Weil cohomology theory by requiring the comparison isomorphism to preserve some 
%or all of this extra structure. We choose to make the following definition. 
%
%A {\em $\Q$-form of the $\ell$-adic 
%cohomology theory $\Hl$} is a $\Q$-form equipped (and compatible) 
%with a cycle map, i.e. a triple $(H,\tau,\gamma)$ where $(H,\tau)$ is a 
%trivialised $\Q$-form and  $\gamma: \CH\Lra H$ is a natural transformation
%from the Chow group functor $\CH:\VV\lra \Qvect$ such that
%$\CH\tensor\Ql\xra{\gamma\tensor\Ql} H\tensor\Ql\xra{\tau} \Hl$
%is the cycle map $\CH\xra{\gamma_{\Het}}\Hl$ of the $\ell$-adic cohomology theory. 
%
%%The comparison isomorphism between singular and $\ell$-adic cohomology of 
%complex smooth projective varieties 
%% the separated $\C$-schemes of finite type
%%provides an  example of a  $\Z$-form 
%%of the $\ell$-adic cohomology theory $\Het(-,\Zl)$. 
We conjecture that, up to action of $\Aut(\C)$ defined above,  
the comparison isomorphism between singular and $\ell$-adic cohomology of 
%complex smooth projective varieties 
%% the separated $\C$-schemes of finite type
%%provides an  example of a  $\Z$-form 
is the only $\Z$-form  
of the $\ell$-adic cohomology theory $\Het(-,\Zl)$:

\begin{enonce}{Conjecture}[$Z(H_{sing},\Hl)$]\label{conj:ZHet} Up to $Aut(\C)$ action, there is a unique  $\Z$-form of the $\ell$-adic
cohomology theory functor $\Het(-,\Zl)$ restricted to the category 
of separated $\C$-schemes of finite type %complex projective smooth algebraic varieties.
which respects the cycle map and Kunneth decomposition.

In other words, every comparison isomorphism of a $\Z$- and the $\ell$-adic cohomology theory of 
 separated $\C$-schemes of finite type
is, up to a field automorphism of $\C$,  the standard comparison isomorphism 
$$H_{sing}(X(\C),\Z)\tensor \Zl = \Het(X,\Zl)
$$
\end{enonce}

The conjecture is intended to be too optimistic; it is probably more reasonable to conjecture uniqueness of $\Z$-form of 
the {\em torsion-free} part of the $\ell$-adic cohomology. 

%Assuming Grothendieck's Standard Conjectures and that, in particular, the $\ell$-adic cohomology factor
%via the category of pure motives, one can state a motivic form of this conjecture.
%In this form, the conjecutre seems to be closed related to the conjectural theory
%of motivic Galois group [Serre], and, quite possible, is implied by the conjectures
%in [Serre] on $\ell$-adic Galois representations and motivic Galois group.

Assume  Grothendieck Standard Conjectures and that, in particular %(?), 
the $\ell$-adic cohomology theory factors via the category $\Mot_k$ of pure motives
over a field $k$. 
Then, a Weil cohomology theory (cf.~[Kleiman]) corresponds to a tensor fibre functor 
from the category of pure motives, and we may ask how many 
 $\Z$-forms does have the fibre functor corresponding to the $\ell$-adic cohomology theory.
Moreover, we may formulate a ``local'' version of the conjecture restricting
the functor to a subcategory generated by a single motive.


\begin{enonce}{Conjecture}[$Z(\Het,\genn E k)$]\label{conj:Zmot} Let $k$ be a number field. 
Assume Grothendieck Standard Conjectures and that, in particular, 
$\ell$-adic cohomology factors via the category of pure numerical motives $\Mot_k$
over $k$. 

Let $E$ be a motive and let $\gen E$ be the subcategory of $\Mot_k$ 
generated by $E$,~i.e.~the 
least Tannakian subcategory of $\Mot_k$ containing $E$. 
Up to $\Aut(\bar{k}/k)$-action, 
the functor $\Het(-\tensor\Qbar,\Zl):\genn E k \lra\ZlVect$ has at most finitely many  $\Z$-forms.

Moreover, if $E$ has finitely many $\Z$-forms [Serre,10.2?], 
then the functor %$\Het(-\tensor \Qbar, {\Bbb A}^f):\genn E k \lra \Qvect$
$\Het(-\tensor \Qbar, \Zhat ):\genn E k \lra \Zhatvect$
has at most finitely many  $\Z$-forms.
\end{enonce}


\subsection{\label{sec:2.2}An example: an Abelian variety.} Let us give an example of a particular case of the conjecture 
which is easy to prove. 


\begin{propo}\label{propo:2.3} 
Let $A$ be an Abelian variety defined over a number field $k$.
Assume that the Mumford-Tate group of $A$ is the maximal possible, i.e.~the symplectic group 
$MT(A)=GSp_{2g}$ where $\dim A = g$,
and that the image of Galois action on the torsion has finite index in the group $GSp_{2g}(\Zl)$ 
of $\Zl$-points
of the symplectic group.

Then there are at most finitely many  $\Z$-form of the $\ell$-adic
cohomology theory $\Het(-\tensor \bar k,\Zl)$ restricted to  the category $\genn A k$, up to $\Aut(\kbar/k)$.
\end{propo}



%Remark/Expectation:
%    If H is a Q-form of H_et, then \Aut(H) is a pro-algebraic group
%    over Q, and \Aut(H) \tensor_\Q \Q_l ~= \Aut(H_et).
%
%    Perhaps some sort of converse also holds?
%
%Example:
%    Conjecture A holds for $A$ an Abelian variety, $\dim A=g$, such that the
%    Mumford-Tate group is $\Symp_{2g}$.
%Proof (sketch):
{\bf Proof} (sketch).
The Weil pairing corresponds %(?) 
 to the divisor corresponding to an ample line bundle over $A$,  
    and by compatibility with the cycle class map of a $\Z$-form and $\Hl=\Het(-\tensor \bar k,\Zl)$
%functoriality we have that, 
%    for a $\Q$-form $H$ of $\Het |_{<A>}$,
    the non-degenerate Weil pairing
%        $\omega : (\Het^1(A))^2 -> \Het^0(A) ~= \Q_l$
$$\omega : (\Hetl^1(A))^2 \lra \Hetl^0(A) = \Z_l$$
restricts to a pairing
%        $\omega : (H^1(A)^2 \lra H^0(A) \approx \Q$,
 $$\omega : (H^1(A))^2 \lra H^0(A) \approx \Z,$$
which is easily seen to be non-degenerate.

    Now let $H_i$ be a $\Q$-form for $i=1,2$.

    Let $(x_i^1,...x_i^g,y_i^1,...,y_i^g)$ be a symplectic basis for $H_i^1(A)$.
    Then each is also a symplectic basis for $\Het^1(A)$,
    and so some $\sigma \in GSp(\Het^1(A),\omega)$ maps $H_1^1(A)$ to $H_2^1(A)$.
    The assumption on the Mumford-Tate group precisely means that such a
    $\sigma$ extends to $\sigma \in \Aut(\Hetl |_{\left<A\right>})$,
    and it follows from the fact that the cohomology of an Abelian variety is
    generated by $H^1$ that $\sigma(H_1) = H_2$.

 Finally, use the assumption on the Galois representation to see 
that there are at most finitely many $\Z$-forms.
\qed %    (TODO: details).



It might be true that a reader familiar with the theory of the motivic Galois group 
may find that a straightforward generalisation of the argument above leads to the following.


\begin{enonce}{Statement}[a generalisation of the example]\label{state:2.4}
%Conjecture (a generalisation of the example):
%   Assume the Mumford-Tate group G=Aut(H_et|<A>)  has the following property:
%if V_1 and V_2 are Q-vector subspaces of H_et(A,Q_l)
%        such that GL(V_i) /\ G(Q_l) is dense in G(Q_l) for i=1,2,
%        then there is a g in G(Q_l) such that gV_1=V_2 (setwise).
%
%    Then A  holds for H_et|<A>.

 Let $A$ be a motive of a smooth projective variety defined over a number field $k$. 
 Assume the Mumford-Tate group $G=Aut^\tensor(\Het(-\tensor\bar k, \Zl)_{|\left<A\right>})$
has the following property: 
 \begin{itemize}
 \item[] if $V_1$ and $V_2$ are abelian subgroups of %$\Q$-vector subspaces of 
$\Het(A\tensor\kbar,\Z_l)$
         which are both dense and of the same rank and 
         such that 
	 \begin{itemize}\item[]$GL(V_i) \cap G(\Z_l)$ is dense in $G(\Z_l)$ for i=1,2,
	 \end{itemize}
        then there is a $g \in G(\Z_l)$ such that $gV_1=V_2$ (setwise).
 \end{itemize}
  Then the conjectures  [2.1?,3.1?,3.2?,9.1?] of [Serre] imply that
 there are at most finitely many  $\Z$-forms of 
$\Het(-\tensor \bar k,\Zl)_{|\left<A\right>}$.
%$\Hetl_{|\left<A\right>\tensor\Qbar}$.
   %  Then $E_{weak}(\Q_l,\left<A\right>\tensor\Qbar)$  holds.
 \end{enonce}


Conjectures  [2.1?,3.1?,3.2?,9.1?] have analogues the cohomology theories
with coefficients in the ring of  finite adeles ${\Bbb A}^f$,
  cf. [Serre, 11.4?(ii), 11.5?], cf. also [Serre, 10.2?, 10.6?].
%We suggest they may be interpreted as
%$E({\Bbb A}^f, \left<E\right>\tensor_k\Qbar)$ 
%where ${\Bbb A}^f$ is the ring of finite adeles.


\subsection{\label{serre}Standard Conjectures and motivic Galois group}

Now we try to give a self-contained exposition of several conjectures 
on motivic Galois group which appear related to our conjectures. 
%place the conjectures above in the context
%of Standard Conjectures and the conjectures
%on the motivic Galois group [Kleiman, Serre].
Our exposition follows [Serre,\S1,\S3]


Let $k$ be a field of characteristic 0 which embeds into the field $\C$ of complex numbers; pick an embedding $\sigma:k\lra\C$. 

Assume Standard Conjectures and Hodge conjecture [Grothendieck, Kleiman]. 
Let $\Mot_k$ denote the category of pure motives over $k$ 
defined with the help of numerical equivalence of algebraic cycles
(or the homological equivalence, which should be the same 
by Standard Conjectures). $\Mot$ is a semi-simple category. 


Let $E\in Ob \Mot$ be a motive; let $\left<E\right>$ denote
the least Tannakian subcategory of $\Mot_k$ containing $E$.

A choice of embedding $\sigma:k\lra\C$ defines 
an exact {\em fibre functor} 
$\Mot_k\lra\QVect$ corresponding to {\em the Betti realisation} 
$${\Hsing}_\sigma: \Mot_k\lra \QVect,\ \ \ E\mapsto \Hsing(E_\sigma(\C),\Q).$$ 

The scheme of automorphisms 
$MGal_{k,\sigma}=\Aut^\tensor({\Hsing}_\sigma:\Mot\lra\QVect)$ 
of the functor preserving the tensor product
is called {\em motivic Galois group of $k$}.
It is a linear proalgebraic group defined over $\Q$. 
Its category of $\Q$-linear representations is equivalent to $\Mot$. 
The group depends on the choice of $\sigma$. 

The motivic Galois group of a motive $E$ is 
$\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)$. 

We now list several conjectures from [Serre]. 

\begin{enonce*}{Conjecture}[2.1?]%\begin{enonce*}{Conjecture}[2.1?] 
The group $\Aut^\tensor({\Hsing}_\sigma:\Mot_k\lra\QVect)$ is proreductive, i.e. a limit of liner reductive $\Q$-groups.
%\end{enonce*}
\end{enonce*}

Let $\boldsymbol 1$ denote the trivial morphism of rank 1, i.e. the cohomology 
of the point $Spec\, k$. 

\begin{enonce*}{Conjecture}[3.1?] The group
$\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)$ is the subgroup of
$GL({\Hsing}_\sigma(E))$ 
preserving the tensors corresponding to morphisms
 ${\boldsymbol 1}\lra E^{\tensor r} \tensor E^{\vee\tensor s}$, $r,s\geq 0$. 
\end{enonce*}

It is also conjectured that this group is reductive. 
Via the comparison isomorphism of \'etale and singular cohomology, 
$$\Hsing( E(\C), \Q ) \tensor \Q_l = \Het ( E\tensor\Qbar, \Q_l ),$$
the $\Ql$-points of $\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql)$ act on the \'etale cohomology $\Het ( E\tensor\Qbar, \Q_l )$.
On the other hand, the Galois group $Gal(\Qbar/k)$ acts
on $\Qbar$ and therefore on $E\tensor\Qbar$. 
By functoriality, the Galois group acts by automorphisms of the functor of \'etale cohomology. 
 Hence, this gives rise to {\em $\ell$-adic representation associated to $E$}
  $$\rho_{k,l}: Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql).$$




\begin{enonce*}{Conjecture}[3.2?] Let $k$ be a number field. 
The image
of the $\ell$-adic representation associated with $E$
is dense in the group $\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql)$ 
in the Zariski topology. 
\end{enonce*}


\begin{enonce*}{Conjecture}[9.1?] Let $k$ be a number field. 
The image $$
Im(\rho_{k,l}: 
Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql))$$ 
is open in $\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql)$.
\end{enonce*}


\begin{enonce*}{Conjecture}[9.3?] Let $k$ be a number field. 
$ \Het ( E\tensor\Qbar, \Q_l )$ is semi-simple as a $Gal(\Qbar/k)$-module. 
\end{enonce*}

We suggest that the  conjectures [2.1?,3.1?,3.2?,9.1?,10.2?,10.3?.10.4?,10.7?,10.8?]
may be interpreted as saying there are only finitely
many $\Z$-forms of the \'etale cohomology 
$\Het(-,\Zl): \left<E\right>\tensor_k\Qbar\lra \QVect$,
up to Galois action. 
%%
%%The following conjecture is probably easy to prove. 
%%
%%\begin{enonce*}{Conjecture}[$E(\Ql, \left<E\right>\tensor_k\Qbar)$] 
%%Let $k$ be a number field and let $E$ be a motive over $k$. 
%%Assume that the action of the motivic Galois group $G=MGal(E)$ of $E$ 
%%on \'etale cohomology satisfies the following condition. 
%% \begin{itemize}
%%  \item[] if $V_1$ and $V_2$ are $\Q$-vector subspaces of $\Het(E\tensor\Qbar,\Q_l)$
%%   which are both dense and of the same dimension and
%%       such that
%%        \begin{itemize}\item[]$GL(V_i) \cap G(\Q_l)$ is dense in $G(\Q_l)$ for i=1,2,
%%            \end{itemize}
%%      then there is a $g \in G(\Q_l)$ such that $gV_1=V_2$ (setwise).
%%  \end{itemize}
%%
%%
%%Then the conjectures [2.1?,3.1?,3.2?,9.1?] above imply that 
%%there are finitely many $\Q$-forms of the \'etale cohomology functor
%%$\Het(-,\Ql): \left<E\right>\tensor_k\Qbar\lra \QVect$. 
%%\end{enonce*}
%%
 There are similar conjectures for finite adeles instead of $\Q_l$,
  cf.~[Serre, 11.4?(ii), 11.5?], also [Serre, 10.2?, 10.6?].
%We suggest they may be interpreted as
%$E({\Bbb A}^f, \left<E\right>\tensor_k\Qbar)$ 
%where ${\Bbb A}^f$ is the ring of finite adeles.

\subsection{\label{sec:2.3}Speculations and remarks.}


Standard conjectures claim there are algebraic cycles corresponding to various cohomological constructions. Model-theoretically it should mean
that something is definable in $ACF$ and it is natural to expect
that such properties be useful in a proof of categoricity, i.e.~in the characterisation of the $\Q$-forms of \'etale cohomology theory. 

We wish to specifically point out the conjectures and properties 
involving smooth hyperplane sections, namely
{\em weak and strong Lefschetz theorems} and {\em Lefschetz Standard Conjecture}, cf.~[Kleiman,p.11,p.14]. 
Weak  Lefschetz theorem describes 
part of the cohomology ring of a smooth hyperplane section of a variety. Perhaps such a description can be useful in showing 
that a $\Q$-form extends uniquely to $Mot/K$ from the subcategory $\Mot/\Qbar$. 
An analogue of the weak Lefschetz theorem for the fundamental group was used in a similar way in [GavrDPhil, Lemma V.III.3.2.1],
see \ref{lefshetz:fund} for some details.
Namely, as is well-known, the fundamental group of a smooth hyperplane section of a smooth projective variety is  
essentially determined by the fundamental group of the variety. 
 [GavrDPhil, III.2.2] extends this to a somewhat technical weaker statement about arbitrary generic hyperplane sections.
An arbitrary variety can be represented as a generic hyperplane section of a variety defined over $\Qbar$ 
and this implies that, in some sense, the fundamental groupoid functor on the subcategory of varieties defined over $\Qbar$
``defines'' its extension to varieties defined over larger fields. 
The word ``defines'' is used in a meaning similar to model theoretic meaning of one first-order language definable in another. 



\begin{question} Find a characterisation 
of the following families of functors:
% H_top ( X(K_\tau), Q ) : Var/K ---> Q-vect
% H ( X(K_tau),C) : Var/K ---> Q-Hodge
 $$\Hsing( X(K_\tau), \Q ) : \Var/K \lra \Qvect,$$
  $$\Hsing( X(K_\tau), \C ) : \Var/K \lra \QHodge$$
where $\tau$ varies though isomorphisms of $K$ to $\C$,
or, almost equivalently, though locally compact locally 
connected topologies on $K$.
\end{question}

Note that Zilber [Zilber] {\em unconditionally} constructs a pseudo-exponential map $ex:\C^+\lra\C^*$ which satisfies the Schanuel conjecture. Of course, this map is not continuous (not even  measurable). 
 Hence we ask:
\begin{question} Construct a pseudo-singular cohomology theory 
which satisfies an analogue of the Schanuel conjecture and some other conjectures.
\end{question}


\subsection{Model theoretic conjectures}

Define model theoretic structures corresponding to the cohomology 
theories.


\begin{enonce}{Conjecture}
   The field is purely embedded into the structures
   corresponding to functors
\begin{itemize}
   \item[(i)]     $\Hsing:Var/\Qbar \lra \QVect$
   \item[(ii)]    $\Hsing(-,\Q):Var/\C \lra \QVect$
   \item[(iii)]   $\Hsing(-,\C):Var/\C \lra \QHodge$
\end{itemize}
Moreover, the structure (ii) is an elementary extension of (i) and
the cohomology ring $\Hsing(V,Q)$ is definable for every variety over C.
\end{enonce}


Several of the Standard Conjectures [Kleiman, \S4,p.11/9] claim that 
certain cohomological cycles (construction) correspond to algebraic 
cycles. This feels related to many of the conjectures above, 
in particular to the purity conjectures.

\begin{problem}
\begin{itemize}
\item[1.] Define a model-theoretic structure and language corresponding
 to the notion of a Weil cohomology theory, and formulate 
 a categoricity conjecture hopefully related to the Standard Conjectures ([Grothendieck, Kleiman])
 and conjectures on the motivic Galois Group and related Galois representations [Serre].

 \item[2.] Do the same in the language of functors, namely:
  \begin{itemize}
  \item[2.1.] Consider the family of cohomology theories on 
  $\Var/$K coming  from a choice of isomorphism $K\approx\C$.
  \item[2.2.] Define a notion of isomorphism of these/such cohomology theories,
    and  what it means to a "purely algebraic" property of such a theory.
   \item[3.3.] Find a characterisation of that family up to that notion of isomorphism
   by such properties. Or rather, show existence of such a characterisation is equivalent to a number of well-known conjectures
   such as the Standard Conjectures etc.
\end{itemize}
\end{itemize}
\end{problem}




\section{\label{pitop:uniq}Uniqueness properties of the topological fundamental groupoid functor of a complex algebraic variety}





\subsection{\label{sec:3.1}Statement of the conjectures}

Let $\VV$ be a category of varieties over a field $K$, let $\pi$
be a functor to groupoids such that $\Points \pi(X)=X(K)$
is the functor of $K$-points.
For $\sigma\in\Aut(K)$, define $\sigma(\pi)$ by
$$\Points\sigma(\pi)=\Points \pi(X) = X(K), \ \ \ 
\sigma(\Paths(x,y))=\Paths(\sigma(x),\sigma(y)),$$ 
$$ \ \ \ {\rm source}(\gamma)=\sigma({\rm source}(\gamma)), {\rm target}(\gamma)=\sigma({\rm target}(\gamma)),$$


For $K=\C$, an example of such a functor is the topological fundamental groupoid functor
$\pi_1^{top}(X(\C))$ of the topological space of complex points of an algebraic variety, 
and $\{\sigma(\pi_1^{top})\,:\,\sigma\in\Aut(\C)\}$ is the family of
all the topological fundamental groupoid functors associated with different choices
of a locally compact locally connected topology on $\C$. (Such a topology determines
a field automorphism, uniquely up to conjugation).

$Aut(K)$ acts by automorphisms of the source category, hence  all these 
(possibly non-equivalent!) % (not necessarily equivalent!)
 functors have the same properties in the language of functors, in particular
\begin{itemize}
\item[(0)] $\Ob\,\pi(X)=X(K)$ is the functor of $K$-points of an algebraic variety $X$
\item[(1)] preserve finite limits, i.e.~$\pi(X\times Y)=\pi(X)\times \pi(Y)$
\item[(2)] $\pi(X)$ is connected if $X$ is geometrically connected (i.e. the set of points $X(K)$ equipped 
with Zariski topology is a connected topological space)
\item[(3)] for $\tilde X\xra f X$ \'etale, the map $\pi(\tilde X)\xra{\pi(f)}\pi(X)$ of groupoids
		has the path lifting property of topological covering maps, namely
          \begin{itemize} 
 	  \item[] for $x =f(\tilde x),\tilde x\in \tilde X(K)$, for every path $\gamma\in \pi(X)$ starting at $x$,
                there is a unique path $\tilde\gamma\in \pi(\tilde X)$ such that %starting at $\tilde x$ such that 
                 ${\rm source}(\gamma)=\tilde x$ and $(\pi(f))(\tilde\gamma)=\gamma$.
          \end{itemize}
\item [(4)] whenever $Y$ is connected, $f:  X\lra Y$ is a proper and separable morphism with
geometrically connected fibres satisfying assumptions of Corollary 1.4 [SGA1,
X], there is a short exact sequence

$pi_1( X_x , x, x) \lra pi_1(X, x, x) \lra \pi_1(Y, f( x), f( x))$
where $X_x$ is the geometric fibre over a geometric point $x$ of $X$.


\end{itemize}


%%{\tt \small >fixme?: use $\tilde f$ or $f_*$ instead of $\pi(f)$, as it is a map of covering spaces, kind of.

%%%>fixme?: introduce a term for the property of $\pi_1^{top}$ being "universal in a family of functors up to a field automorphism"?  }


\begin{defi}{\em A $\pi_1$-like functor over a field $K$} is a functor %$\pi:\Var_K\lra\Groupoids$ 
from a category of varieties to the category of groupoids
satisfying (0-4) above. 
A $\pi_1$-like functor is a $\pi_1$-like functor over some field.
\end{defi}
Note that by (0) a $\pi_1$-functor comes 
equipped with a forgetful natural transformation to the functor of $K$-points. 



\begin{enonce}{Conjecture}[$Z(\pi_1^{top},\Var_\C)$]\label{conj:Zpitop} Each $\pi_1$-like functor on the category of smooth quasi-projective complex varieties
factors through the topological fundamental groupoid functor, up to a field automorphism. 

In detail: Let $\Var_\C$ be the category of smooth quasi-projective varieties over the field of complex numbers $\C$.
For each $\pi_1$-like functor $\pi:\Var_\C\lra\Groupoids$ there is a field automorphism $\sigma:\C\lra\C$ and a natural
transformation $\varepsilon:\pi_1^{top}\Lra\pi^\sigma$ such that the induced natural transformation $\Ob\pi_1^{top}\Lra\Ob\pi^\sigma$
on the functor of $\C$-points 
is identity.  
\end{enonce}



\begin{enonce}{Conjecture}[$Z(\pi_1,K,\Var_K)$]\label{conj:ZpiK}
Let $K$ be an algebraically closed field.  Let $\Var_K$ be the category of smooth quasi-projective varieties over $K$.

There is a functor $\pi_1:\Var_K\lra\Groupoids$ such that 
for each $\pi_1$-like functor $\pi:\Var_K\lra\Groupoids$ there is a field automorphism $\sigma:K\lra K$ and a natural
transformation $\varepsilon:\pi_1\Lra \pi^\sigma$ such that the induced natural transformation $\Ob\pi_1\Lra\Ob\pi^\sigma$
on the functor of $K$-points 
is identity.  
\end{enonce}


These conjectures are direct analogues of categoricity conjectures in model theory,
hence we shall refer to these conjectures as {\em categoricity conjectures for the fundamental
groupoid functor}.

\begin{rema} As stated, these conjectures are likely too optimistic. To get more plausible and manageable conjectures,
replace $\Var_K$ by a smaller category and add additional conditions on the $\pi_1$-like functors. 
The conclusion can also be weakened to claim there is a finite %pseudo-versal 
family of functors, rather than a single functor, through which $\pi_1$-like functors factor
up to field automorphism.

It may also be necessary to put extra structure on the fundamental groupoids. 
\end{rema}

\begin{rema} In model theory, it is more convenient to work with universal covering spaces 
rather than fundamental groupoids. Accordingly, model theoretic results are stated in the language
of universal covering spaces, sometimes with extra structure. 

 The conjectures above are motivated by questions and theorems about categoricity of certain structures. 
\end{rema}


\def\sepp{\mathrm{sep}}\def\Spec{\mathrm{Spec}\,}
\begin{rema}\label{rem:autpi} It is tempting to think that the right generalisation of the conjectures above should 
make use of the short exact sequence of \'etale fundamental groups (see [SGA1, XIII.4.3;XII.4.4])
$$1\lra 
\pi_1^{alg}(X\times_{\mathrm{Spec} k} \mathrm{Spec}k^\sepp, x)  \lra  \pi_1^{alg}(X, \bar x) 
\lra \pi_1(\Spec k, \mathrm{Spec}k^\sepp) =\Gal(k^\sepp/k)\lra 1$$
where $X$ is a scheme over a field $k$, $k^\sepp$ is a separable closure of $k$, and  $x:\Spec k^\sepp \lra X\times_{\mathrm{Spec} k} \mathrm{Spec}k^\sepp$ is a geometric point of $X\times_{\mathrm{Spec} k} \mathrm{Spec}k^\sepp$,
and $\bar x : \Spec k^\sepp \lra X\times_{\mathrm{Spec} k} \mathrm{Spec}k^\sepp\lra X $ is the corresponding geometric point of $X$.
 
In fact such a sequence could be associated with a morphism $X\lra S$ 
admitting a section and satisfying certain assumptions [SGA 1, XIII.4].

These short exact sequences comes from pullback squares
\def\pullbacksquare#1#2#3#4#5#6{\xymatrix{ {#1} \ar[r]|{} \ar@{->}[d]|{#2} & {#4} \ar[d]|{#5} \\ {#3}  \ar[r] & {#6} }}
$$\pullbacksquare{X\times_{\mathrm{Spec} k} \mathrm{Spec}k^\sepp}{}{X}{\Spec k^\sepp}{}{\Spec k}
\ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ 
\pullbacksquare{X_s}{}{X}{s}{}{S}
$$

\end{rema}

\subsection{\label{sec:3.2}Conjectures within reach.} In this section we give two conjectures which
we hope to be within reach. Their proof requires a combination of methods of model
theory and algebraic geometry. In the next subsection we list several partial positive
results towards conjectures above which are implicit in model theoretic literature.


We find the following conjecture plausible and hope its statement clarifies the arithmetic nature of our conjectures.
It is perhaps the simplest conjecture not amendable to model theoretic analysis because it uses bundles,
a notion from geometry rather than model theory. %(>fixme?: weird wording).


For a variety $X$, let $\genn X K$ denote the category  whose objects are the finite Cartesian powers 
of $X$, and morphisms are morphisms of algebraic varieties defined over $K$; 
we let $X^0$ denote a variety consisting of a single $K$-rational point.  



\begin{enonce}{Conjecture}[$Z(\pi_1,L_A^*)$] Let $K$ be an algebraically closed field of zero characteristic, 
$A$ an Abelian variety defined over a number field $k$. Let $L_A$ be an ample line bundle over $A$
 and $L_A^*$ be the corresponding ${\Bbb G}_m$-bundle. 
Further assume that the Mumford-Tate group of $A$ is the maximal possible, i.e.~the general symplectic group,
$$MT(A)=\GSpZ$$
and that the image of Galois action on the torsion has finite index in the group of ${\Bbb {\hat{Z}}}$-points
of the symplectic group.

Then there is a finite family of $\pi_1$-like functors $\Pi_1$ such that each $\pi_1$-like functor 
 on the full subcategory $\genn {L_A^*} K$ %$\VV_{L_A^*}$, $\Ob \VV_{L_A^*} = \{(L_A^*)^n\,:\,n\geq 0\}$
consisting of the Cartesian powers of the ${\Bbb G}_m$-bundle $L_A^*$,
factors via an element of $\Pi_1$.
\end{enonce}

These functors in $\Pi_1$ correspond to different embeddings of the field of definition of $A$ into the field of complex numbers. 

The following conjecture is probably within reach, at least if we replace the fundamental groupoid functor by its residually finite part. 

Model theoretic methods of $[\mathrm{BH^2K^2 14}]$ 
are likely enough 
to show that it is enough
to prove this conjecture for
a countable algebraically closed subfield. 
Methods of [GavrDPhil,III.5.4.7], %cf.~\S\ref{results}.\ref{w-cat} and 
cf.~\S\ref{lefshetz:fund} and particularly Statement~\ref{w-cat},  
are likely enough to reduce the remaining part of the conjecture 
to certain properties of complex analytic topology
and normalisation of varieties.



\begin{enonce}{Conjecture}[$Z(\pi_1,\Qbar\subset \C, \Var_{\Qbar}\subset \Var_\C)$]\label{conj:ZVqVc}
%Let $K$ be an algebraically closed field extending $\Qbar$.  
Let $\Var_\C$ be the category of smooth quasi-projective varieties over $\C$,
and let $\Var_{\Qbar}$ be its subcategory consisting of varieties and morphisms defined over $\Qbar$.

Assume that $\pi:\Var_\C\lra \Groupoids$ is a $\pi_1$-like functor 
which coincides with the topological fundamental groupoid 
$\pi_1^{top}$ for varieties and morphisms defined over $\Qbar$, 
i.e.~for each variety $V$ in $\Var_{\Qbar}$ %defined over $\Qbar$ 
and each morphism $V\xra f W$ in $\Var_{\Qbar}$ % defined over $\Qbar$ 
 it holds
$\pi(V)=\pi_1^{top}(V)$ and $\pi(f)=\pi_1^{top}(f)$. 

Then there exist a field automorphism $\sigma\in \Aut(\C/\Qbar)$
such that $\pi\circ\sigma$ and $\pi_1^{top}$ are equivalent.
\end{enonce}


There are a number of theorems and conjectures which can be seen as saying that, 
up to finite index, the Galois action is described by geometric, algebraic or topological structures;
our conjectures can also be seen in this way. 


%These conjectures are probably overly optimistic. More plausible conjectures
%are obtained by considering a smaller family of functors and a smaller source category 
%which is large %or self-contained? 
%enough to capture relevant restrictions on the Galois action. 
%The conclusion can also be weakened to claim there is a finite %pseudo-versal 
%family of functors, rather than a single functor, through which $\pi_1$-like functors factor
%up to field automorphism.
%


\subsection{\label{results}Partial positive results} These conjectures are closely related to {\em categoricity} 
theorems in model theory, and this has led to several partial positive results 
uniqueness of $\pi_1$-like functor restricted to certain full subcategories of the category of varieties
over an algebraically closed field:
the full subcategories $\gen {K^*}$ of algebraic tori in arbitrary characteristic,
$\gen E$ powers of an elliptic curve over a number field, a weaker result about
$\gen A$ powers of an Abelian variety over a number field, a still  weaker result
about $\gen V$ powers of a smooth projective variety whose fundamental group 
satisfies a group theoretic property of being subgroup separable,
a strengthening of residually finite. 

Note that the first three categories are linear in the sense that the the groups $\Aut(K^*)$,
$\Aut_{End\,E-mod}(E(K))$, and $\Aut_{End\,A-mod}A(K)$ act on the set of $\pi_1$-like functors on 
the respective categories $\gen {K^*}$, $\genn E K$, and $\genn A K$. This is so because
these groups act on these categories.


Let us now list several statements translated from categoricity theorems available
in model theory literature. We give references but do not explain the translation.


A reader familiar with algebraic geometry may find that there is a better way to
formulate these results. In the case of a countable field $K$, their proofs do not require
elaborate techniques of model theory, and such a reader may find it easier to reprove
these results in a familiar language while only drawing inspiration from the proofs in
the literature.





For a variety $V$ over a field $K$, let $\genn V K$ denote the full subcategory of the category of 
varieties defined over $K$ whose objects are all finite Cartesian powers of $V$
and a $K$-rational point. 


Statement~\ref{ref:Gm} is a particular case of  Conjecture~\ref{conj:ZpiK}($Z(\pi_1,K,\Var_K)$) 
when we take $\Var_K=\genn {K^*} K$ to be the category of algebraic tori. Note this implies 
 Conjecture~\ref{conj:Zpitop}($Z(\pi_1^{top},\genn {\C^*} {\C} )$).

\begin{state}[BaysZilber,Th.2.1;GavrK] \label{ref:Gm} Let $K$ be an algebraically closed field of $\char K=0$. 
Then there is a $\pi_1 $-like functor $\pi$ over $K$ defined on the category $\genn {K^*} K$ of algebraic tori
 such that for any $\pi_1 $-like functor $\pi'$ over $K$ defined on the same category 
there is a field automorphism   $\sigma:K\lra K$ and a natural transformation 
$\pi \Longrightarrow \sigma(\pi')$ 
such that the induced natural transformation $Ob \pi \Longrightarrow   Ob \sigma(\pi')$  is identity.
\end{state}

Similarly, 
Statement~\ref{ref:EK} is a particular case of  Conjecture~\ref{conj:ZpiK}($Z(\pi_1,K,\genn E K)$) 
when we take $\Var_K=\genn {E} K$ to be the category generated by an elliptic curve defined over a number field, 
and weaken the conclusion to require finitely many rather than unique. This is necessary, as there are 
finitely many embedding of the number field into the field of complex numbers, and they give rise 
to non-equivalent $\pi_1$-like functors.  Note this implies 
 Conjecture~\ref{conj:Zpitop}($Z(\pi_1^{top},\genn E \C)$) weakened in a similar way.


\begin{state}[BaysDPhil, Th.4.4.1; GavrK, Prop.2]\label{ref:EK} 
 Let $K$ be an algebraically closed field of $\char K=0$, and 
let $E$ be 
 an elliptic curve defined over a number field $k$
with a $k$-rational point $0\in E(k)$. 

Then there are finitely many $\pi_1$-like functors $\pi^1,...,\pi^n$ on the category $\genn E K$ 
such that for any functor $\pi$ such that
\begin{itemize}
\item $\pi(E,0,0)\approx \Z^2$
\end{itemize}
there is a field automorphism   $\sigma:K\lra K$ and a natural transformation 
$\pi \Longrightarrow \sigma(\pi^i)$ for some $i$  
such that the induced natural transformation $Ob \pi \Longrightarrow   Ob \sigma(\pi^i)$  is identity.
\end{state}

% and $\VV$, $\Ob\VV=\{E^n\,:\,n\geq 0\}$ is the full subcategory
%  finite powers of an elliptic curve $E^n$, $n\geq 0$ defined over a number field.
% Then there are finitely many functors $\pi^1,...,\pi^n$ in the family such that
%every other functor factors via one of them, up to a field automorphism.


Statement~\ref{ref:GmFp} is a particular case of 
Conjecture~\ref{conj:ZpiK}($Z(\pi_1,K,\genn E K)$) 
when we take $K$ to be an algebraically closed field of prime characteristic,  
$\Var_K=\genn {K^*} K$ to be the category of algebraic tori,
and weaken the conclusion by placing additional restrictions on $\pi_1$-functors. 

These restrictions are necessary because there are too few automorphisms of the the algebraic closure $\bar{\Bbb F}_p$
of the field with $p$ elements. However, we may view this as a hint towards Remark~\ref{rem:autpi}
suggesting we should replace below $\Aut(K/\Fpbar)$ by the etale fundamental group of an appropriate scheme over $\Bbb F_p$.




\begin{state}[BaysZilber,Th.2.2]\label{ref:GmFp} %$\char K=p>0$, $\gen {K^*}$; 
%there is a pseudo-versal functor up to  $\Aut(K/\Fpbar)$ 
%over the functor of $K$-points for any family $\mathcal F$ of fundamental groupoid-like functors
 Let $K$ be an algebraically closed field of $\char K=p$. 
Then there is a $\pi_1$-like functor $\pi_1:\gen {K^*}\lra \Groupoids$ such that each $\pi_1$-like functor
$\pi:\gen {K^*}\lra \Groupoids$ factors via $\pi_1:\gen {K^*}\lra \Groupoids$ up to $\Aut(K/\Fpbar)$
provided
\begin{itemize}
 \item $\pi(K^*,1,1)\approx {\Bbb Z}[1/p]$  
\item  %for any two functors $\pi,\pi'$ in $\mathcal F$,  
the restrictions  ${\pi_1}_{|\bar{\Bbb F}_p}$ and $\pi_{|\bar{\Bbb F}_p}$ 
to $\Fpbar$-rational points coincide:
$$\pi_{|\bar{\Bbb F}_p}=\pi'_{|\bar{\Bbb F}_p}$$
 \end{itemize} 
\end{state}



Statement~\ref{ab:cat} is a particular case of 
Conjecture~\ref{conj:ZpiK}($Z(\pi_1,K,\genn A K)$),  
for $A$ is an abelian variety over a number field.
weakened by requiring the fundamental {\em group} (rather than groupoid) 
functors of the $\pi_1$-like functors 
to coincide.



\begin{state}[BaysDPhil, Th.4.4.1] \label{ab:cat} %$\char K=0$, $\genn A K$
%$\VV$ is the full subcategory whose objects are $\Ob\VV=\{A^n\,:\, n\geq 0\}$ 
 Let $K$ be an algebraically closed field of $\char K=0$. Let 
 $A$ be an Abelian variety defined over a number field $k$ with a $k$-rational point $0\in A(k)$.

Then there is a $\pi_1$-like functor $\pi_1:\gen {A}\lra \Groupoids$ such that each $\pi_1$-like functor
$\pi:\gen {A}\lra \Groupoids$ factors via $\pi_1:\gen {K^*}\lra \Groupoids$ up to $\Aut(K/\Q(A_{tors}))$
whenever
 \begin{itemize}
 \item $\pi(A,0,0)={\Bbb Z}^{2\dim A}$
%%%% \item  for any two functors $\pi,\pi'$ in $\mathcal F$,  
%%%%  the corresponding fundamental group functors coincide
%their restrictions  $\pi_{|{\Bbb F}_p^*}$ and $\pi'_{|{\Bbb F}_p^*}$                   to $\Fpbar$-rational points coincide:
%$$\pi_{|{\Bbb F}_p^*}=\pi'_{|{\Bbb F}_p^*}$$
 \item  for any two functors $\pi,\pi'$ in $\mathcal F$,  
the corresponding fundamental group functors coincide
$$\pi(A,0,0)=\pi'(A,0,0)$$
and further, for $p:\tilde A\lra A$ is \'etale, $\gamma\in\pi(A,0,0)=\pi'(A,0,0)$,
$\pi(\gamma_\pi)=\pi'(\gamma_{\pi'})=\gamma$, it holds that
$${\rm source}(\gamma_\pi)={\rm source}(\gamma_{\pi'})\ \ \ {\rm 
implies }\ \ \ \ {\rm target}(\gamma_\pi)={\rm target}(\gamma_{\pi'})$$ 
%{\tt >fixme!: check whether the above two items are equivelent}  
%their restrictions  $\pi_{|{\Bbb F}_p^*}$ and $\pi'_{|{\Bbb F}_p^*}$                   to $\Fpbar$-rational points coincide:
%$$\pi_{|{\Bbb F}_p^*}=\pi'_{|{\Bbb F}_p^*}$$
 \end{itemize} 
\end{state}


Statement~\ref{w-cat} is a particular case of Conjecture~\ref{conj:ZVqVc}($Z(\pi_1,\Qbar\subset \C, \Var_{\Qbar}\subset \Var_\C)$)
when $\Var_{\Qbar}=\genn V {\Qbar}$ and $\Var_\C=\genn V {\C}$ are the subcategories generated by a variety with certain properties.



Recall a group $G$ is subgroup separable iff for each finitely generated subgroup $H<G$
and $h\not\in H$ there is a morphism $f:G\lra G_H$ into a finite group $G_H$ such that
$h\not\in f(H)$. 


\begin{state}[GavrDPhil,III.5.4.7] \label{w-cat} %$\char K=0$, ${\rm card\,}K=\aleph_1$, and $\genn V K$
%$\VV$ is the full subcategory whose objects are $\Ob\VV=\{V^n\,:\, n\geq 0\}$ 
 Let $K$ be an algebraically closed field of $\char K=0$ and of the least uncountable cardinality $\aleph_1$.
Let  $V$ be an smooth projective variety defined over a number field $k$
with a $k$-rational point $0\in V(k)$ such that the universal covering space of $V(\C)$ is holomorphically complex, for some embedding $K\hookrightarrow \C$, 
and  its fundamental groups $\pi_1(V(\C),0,0)^n$ are subgroup separable for each $n>0$.

Then there is a $\pi_1$-like functor $\pi_1:\gen V\lra \Groupoids$ such that a $\pi_1$-like functor
$\pi:\gen {V}\lra \Groupoids$ factors via $\pi_1:\gen {V}\lra \Groupoids$ up to $\Aut(K/\Qbar)$
provided
\begin{itemize}
 \item $\pi(V,0,0)\approx \pi_1^{top}(V(\C),0,0)$ 
\item
the restrictions  ${\pi_1}_{|\bar{\Bbb Q}}$ and $\pi_{|\bar{\Bbb Q}}$ 
to $\Qbar$-rational points coincide:
$$\pi_{|\bar{\Bbb Q}}=\pi'_{|\bar{\Bbb Q}}$$
 \end{itemize} 
\end{state}


%there is a finite pseudo-versal family for any family of fundamental groupoid-like functors
%such that
% \begin{itemize}
% \item $\pi(V,0,0)\approx \pi_1^{top}(V(\C),0,0)$ 
% \item for any two functors $\pi,\pi'$ in $\mathcal F$,  
%$$\pi_{|\bar{\Bbb Q}}=\pi'_{|\bar{\Bbb Q}}$$
% \end{itemize} 


We wish to mention the work of [HarrisDPhil, DawHarris] and particularly [Eterovich] on Shimura curves, which does not quite fit 
in our framework. To interpret their results, one needs to consider $\pi_1^{top}$ 
as a functor to groupoids {\em with extra structure}.  



\begin{rema}
The conjectures on independence of Galois representations of non-isogenious curves likely imply our conjectures 
for the full subcategory
$\genn {E_1\times\ldots\times E_n} K$
generated by a finite product of elliptic curves $E_1,\ldots,E_n$ over a number field. 


Consider the family of $\pi_1$-like functors with Abelian fundamental groups.
%This requires weakening of the uniqueness in the path-lifting property (3) 
%in the definition of the $\pi_1$-like functor. ???? 

Is it true that each such ``abelianised'' $\pi_1$-like functor factors via $\pi_1^{top}$ up to a  field automorphisms?
Is this equivalent to Conjecture~\ref{conj:Zpitop}($Z(\pi_1^{top},AbVar)$) for the subcategory $AbVar$ of all abelian varieties? 
\end{rema}

%An analogue of 
%(\ref{ab:cat}) can probably be proven for the Abelianised fundamental groupoid functor, 
%i.e. the family of functors satisfying the additional property that the fundamenthal group is Abelian. 

\subsection{\label{sec:3.4}Mathematical meaning of the conjectures. Elements of proof of the conjectures}

Here we try to explain the arithmetic and geometric meaning of the conjectures. 
In a sense, the conjectures say that $\Gal(\Qbar/\Q)$ and $\Aut(K/\Q)$ are large enough.
We try to show below in what sense, by showing possible obstructions/difficulties in proof.


\subsubsection{Galois action on roots of unity and Kummer theory}
Consider the infinite sequence  $exp(2\pi i/n)$ of roots of unity. This sequence can be obtained
topologically: take the loop $\gamma$ generating $\pi(\C^*,1,1)\approx \Z$, the \'etale 
morphism $z^n:\C^*\lra\C^*$ and lift $\gamma$ uniquely to a path $\tilde\gamma_n$ starting at $1\in \C^*$. 
Then $exp(2\pi/n)$ is the end-point of $\gamma_n$. This construction shows that a $\pi_1$-like functor on 
the category $\gen {K^*}$ determines a distinguished sequence $\xi_n,n\geq 0$ of roots of unity. 
 Hence, our conjectures require that the Galois group acts transitively on the set of sequences of roots of unity
associated with $\pi_1$-like functors. %These sequences form a group we call $\mu_\infty$.  


Consider a $\pi_1$-like functor on the category $\gen {K^*}$. As noted above, group automorphisms $\Aut(K^*)$ 
of $K^*$ act on the set of these functors. Hence, item (\ref{ref:Gm}) requires that multiplicative group automorphisms 
$\Aut(K^*)$ and field automorphisms $\Aut(K/\Q)$ have the same orbits on the sequences $\xi_n,(\xi_{mn})^{m}=\xi_n,m,n>0$ 
of roots of unity.  

Kummer theory arises in a similar way if we consider endpoints of liftings of paths joining $1$ and 
arbitrary elements $a_1,...,a_n$.

\subsubsection{Elliptic curves and Abelian varieties. Kummer theory and Serre's open image theorem
for elliptic curves.}
Kummer theory for elliptic curves and Abelian varieties arises in the same way if we consider 
$\pi_1$-like functors on the category $\gen A$ generated by an Abelian variety. 

 Similarly, our conjectures 
about $\pi_1$-like functors on $\genn A K$ require that the action of $Aut_{EndA{\rm-mod}}(A(K))$
and $\Gal(\Qbar/k)$ on the torsion points do not differ much. This is true for elliptic curves
but fails for Abelian varieties of $\dim A>1$, hence the extra assumption in (4) on the family of $\pi$-like functors. 



\subsubsection{\label{lefshetz:fund}Arbitrary variety. Etale topology and an analogue of Lefshetz theorem for the fundamental group}

To prove Statement~\ref{w-cat}, we need several facts about \'etale topology. Most of these facts are well-known for smooth varieties;
what we use is that they hold ``up to finite index'' for arbitrary (not necessarily smooth or normal) subvarieties
of a smooth projective variety. 

\def\limVC{\varprojlim\tilde V(\C)}
Consider the inverse limit $\varprojlim \tilde V(\C)$ of finite \'etale covers $\tilde V(\C) \lra V(\C)$  of a complex algebraic 
variety $V$. The universal analytic covering map $U\lra V(\C)$ gives rise to covering maps $U\lra \tilde V(\C)$ and hence
a map $U\lra \limVC$. Zariski topology on the \'etale covers makes $\limVC$ into a topological space. Hence 
there are two topologies on $U$ -- the complex analytic topology and the ``more algebraic"  topology
on $U$ induced from the map $U\lra\varprojlim V(\C)$. Call the latter {\em \'etale} topology on $U$.

\begin{defi} The \'etale topology on the universal covering space of the
topological space of complex points of an algebraic variety is defined as the
topology induced from the map to the inverse limit of the spaces of complex
points of finite \'etale
covers of the variety equipped with the Zariski topology. The \'etale topology on a covering
space of the topological space of complex points of an algebraic variety is defined
similarly.
\end{defi}

To prove Statement\ref{w-cat} we use that these two topologies are similar and nicely
related. In particular,

\begin{lemm} Assume $V$ satisfies the assumptions of Statement~\ref{w-cat}.
\begin{itemize}
\item Closed irreducible sets in \'etale topology are closed irreducible in complex analytic topology (by definition).
\item For a set closed in \'etale topology, its irreducible components in complex analytic topology 
  are also closed in \'etale topology [GavrDPhil,III.1.4.1(4,5)].
\item The image of an \'etale closed irreducible subset of $U\times\ldots\times U$ under a coordinate projection is \'etale closed [GavrDPhil, III.2.2.1].
\end{itemize} 
\end{lemm}

Note that this is easy to see that {\em connected} components of a set closed in \'etale topology are also 
closed in \'etale topology, 
and hence that the properties above holds for smooth or normal closed sets. 

Let $f:W\lra V$ be a morphism of varieties, and let $f_*:U_W\lra U_V$ be the map of the universal covering 
spaces of $W(\C)$ and $V(\C)$.  We may assume that $V$ is smooth projective but it is essential that $W$ is
arbitrary. In applications, $W$ is an arbitrary closed subvariety of a Cartesian power of a fixed variety $V$. 

\begin{lemm} Under the assumptions above, %Assume $V$ satisfies the assumptions of Statement~\ref{w-cat}.
%\begin{itemize}
%\item If $Z$ is closed irreducible in \'etale topology on $U_V$, then the image $f(Z)$ is closed irreducible in \'etale topology.
%\item %if $V$ and $W$ are complete, then 
       if $f:W\lra V$ is proper, then the image $f(U_W)$ is closed in $U_V$ in \'etale topology.
%\end{itemize} 
\end{lemm}
This is related to the following well-known geometric fact: % [GavrDPhil, V.3.3.6, V.3.4.1]:
\begin{lemm} 
\begin{itemize}
\item If $f:W(\C)\lra V(\C)$ is a morphism of smooth normal algebraic varieties, $g$ a generic point of $V(\C)$
and $W_g=f^{-1}(g)$ then 
$$
\pi_1(W_g,w,w)\lra\pi_1(W,w,w)\lra \pi_1(V,g,g)$$
is exact up to finite index.
\item Moreover, if $f(W(\C))$ is dense in $V(\C)$, then $\pi_1(W,w,w)\lra \pi_1(V,g,g)$ is surjective.
\end{itemize} 
\end{lemm}
In fact we need to use a generalisation of this, namely that it holds up to finite index for arbitrary varieties
if one considers the image of the fundamental group in the ambient smooth projective variety,
see  [GavrDPhil, V.3.3.6, V.3.4.1] for details.

%%{\tt >fixme!: proofread and explain better}






{\bf Acknowledgement.} Ideas and proofs were strongly influenced by extensive conversations with Martin Bays.
I thank A.Luzgarev and V.Sosnilo for useful discussions. 
I thank Sergei Sinchuk for helpful discussions.
I also thank Maxim Leyenson for comments on a late draft. 
I thank Yves Andre for several corrections. 

 Support from Basic Research Program of the National Research University Higher                                                                             
School of Economics is gratefully acknowledged. This study was partially supported by                                                                          
the grant 16-01-00124-a of Russian Foundation for Basic Research.

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\end{document}

To get more managable conjectures, we may 
(i) consider a smaller subcategory of the projective smooth
algebraic varieties, (ii) claim there are at most finitely many trivialised $\Q$-forms
(iii) add further restrictions on the functors such as assume they both factor 
thru the category of pure motives. 


\end{document}



 
\begin{enonce*}{Conjecture}
Let $\VV$ be the the category of smooth quasi-projective varieties 
over the field of complex numbers. The topological fundamental groupoid 
functor $ \pi_1^{top} $ is pseudo-universal over the functor of $\C$-points
for the family of functors to groupoids satisfying (0-3) 
and the following
somewhat technical requirement:
\begin{itemize}
\item[(4)] if $\gamma\in \Mor\pi(X)$ is not trivial, then there is an \'etale morphism $Y\xra f X$ in $\VV$ 
such that ${\rm source}(\tilde \gamma)\neq {\rm target}(\tilde \gamma)$ and $f(\tilde\gamma)=\gamma$. 
\end{itemize}


Then for each functor $\pi:\VV\lra Groupoids$ satisfying (0-3) 
there is a field automorphism $\sigma:K\lra K$ and 
a natural transformation $\varepsilon: \pi_1^{top} \Lra \sigma(\pi)$ such that
the induced natural transformation $\Points \pi_1^{top} = \Points \pi)$ is identity.
\end{enonce*}




\begin{enumerate}
\item $\char K=0$ and $\VV$, $\Ob\VV=\{(K^*)^n\,:\,n\geq 0\}$ is the full subcategory of algebraic tori $(K^*)^n$, $n\geq 0$

\item  $\char K=0$ and $\VV$, $\Ob\VV=\{E^n\,:\,n\geq 0\}$ is the full subcategory
  finite powers of an elliptic curve $E^n$, $n\geq 0$ defined over a number field.
 Then there are finitely many functors $\pi^1,...,\pi^n$ in the family such that
every other functor factors via one of them, up to a field automorphism.


\item  $\char K=0$ and $\VV$, $\Ob\VV=\{E^n\,:\,n\geq 0\}$ is the full subcategory
  finite powers of an elliptic curve $E^n$, $n\geq 0$ defined over a number field.
 Then there are a finite family of functors pseudo-versal over the functor of 
$K$-point, up to Aut(K), for the family of functors satisfying (0-3).


\item $\char K=p$ and $\VV$ is the full category of algebaric tori $(K^*)^n,n>0$, and, 
for some fixed functor $\pi^0$ in the family, 
 \begin{itemize}
 \item[($3^!$)] $\pi$ has the unique path lifting property
 \item[(4)]   $\pi(K^*,1,1)=\pi^0(K^*,1,1)$ 
\item[($4'$)] $\pi({\Bbb F}_p^*)=\pi^0({\Bbb F}_p^*)$

\item $\pi(K^*)_{|\Fpbar}=\pi'(K^*)_{|\Fpbar}$
 \end{itemize} 



\item \label{ab:cat} $\char K=0$, $\VV$ is the full subcategory whose objects are $\Ob\VV=\{A^n\,:\, n\geq 0\}$ 
 where $A$ is an Abelian variety defined over a number field $k$ with a $k$-rational point $0\in A(k)$, 
and, for some fixed functor $pi^0$ in the family, 
 \begin{itemize}
 \item[($3^!$] $\pi$ has the unique path lifting property
 \item[(4)] $\pi(A,0,0)=\pi^0(A,0,0)$
 \end{itemize} 


\item $\char K=0$, ${\rm card\,}K=\aleph_1$, and $\VV$ is the full subcategory whose objects are $\Ob\VV=\{V^n\,:\, n\geq 0\}$ 
 where $V$ is an smooth projective variety defined over a number field $k$ such that
 the its fundamental group $\pi_1(V,0,0)^n$ is subgroup separable for $n>0$, 
and 
 \begin{itemize}
 \item[(5)] $\pi(V(\bar{\Bbb Q}))=\pi^0(V(\bar{\Bbb Q}))$
 \end{itemize} 

\end{enumerate}





\begin{defn} 
%Let $\FF$ be a family of functors $\pi$ each equipped with a natural transformation 
%$\pi\Lra \pi_0$. Let $G$ be a group
Let $\FF$ be a family of functors $\pi:\VV\lra\Groupoids$ to groupoids %satisfying property (0)
such that $\Points\pi$ is the functor of $K$-points, and let 
$G$ be a group acting on $\FF$ such that $\Points\pi^\sigma=\Points\pi$. 
%fixme: maybe say: "commuting with natural transformation pi==>Points pi"

A functor $\pi_1$ is {\em pseudo-versal for a family $\mathcal F$ of functors up to action $G$ 
over the functor of $K$-points} iff for each functor $\pi\in {\mathcal F}$ there is $\sigma\in G$ 
and a natural transformation $$\pi_1\xra{\varepsilon}\pi^\sigma$$ 
such that the induced natural transformation
$$\Points\pi_1\xra{\varepsilon}\pi^\sigma$$ on the functor of $K$-points is the identity. 

We say  functor $\pi_1$ is {\em pseudo-universal et al} iff such a natural transformation 
$\Points\pi_1\xra{\varepsilon}\pi^\sigma$ is unique.
\end{defn}

>>FIXME??: which way to state the conjecture is the neatest ? 
Do I really need to introduce the definiton of pseudo-versal functor? 

\begin{enonce*}{Conjecture} Let $K=\C$ be the field of complex numbers.
 Up to $Aut(\C/\Q)$ action, the topological fundamental groupoid functor
from the category of smooth quasi-projective varieties 
maps into any functor with properties (0-3) over the functor of $\C$-points. 
\end{enonce*}


\begin{enonce*}{Conjecture} Let $K=\C$ be the field of complex numbers.
 Up to $Aut(\C/\Q)$ action, the topological fundamental groupoid functor
from the category of smooth quasi-projective varieties 
maps into any $\pi_1$-like %fundamental groupoid-like 
functor over the functor of $\C$-points. 
\end{enonce*}




\begin{enonce*}{Conjecture} Let $K=\C$ be the field of complex numbers.
 Up to $Aut(\C/\Q)$ action, the topological fundamental groupoid functor
from the category of smooth quasi-projective varieties
is pseudo-versal over the functor of $\C$-points for the family of fundamental groupoid-like functors. 
\end{enonce*}


\begin{enonce*}{Conjecture} Let $K=\C$ be the field of complex numbers.
 Up to $Aut(\C/\Q)$ action, the topological fundamental groupoid functor
from the category of smooth quasi-projective varieties
is pseudo-versal over the functor of $\C$-points for the family of functors satisfying properties (0-3). 
\end{enonce*}




\begin{enonce*}{Conjecture}
Let $\VV$ be the the category of smooth quasi-projective varieties 
over the field of complex numbers. The topological fundamental groupoid 
functor $ \pi_1^{top} $ is pseudo-versal over the functor of $\C$-points
for the family of functors to groupoids satisfying (0-3). 

That is,  for each functor $\pi:\VV\lra Groupoids$ satisfying (0-3) 
there is a field automorphism $\sigma:\C\lra \C$ and 
a natural transformation $\varepsilon: \pi_1^{top} \Lra \sigma(\pi)$ such that
the induced natural transformation $\Points \pi_1^{top} = \Points \pi)$ is identity.
\end{enonce*}

Neither topology nor zero characteristic is necessary to state the conjecture:

\begin{enonce*}{Conjecture}
Let $K$ be an algebraically closed field and let $\VV$ be the category of smooth quasi-projective
varieties over $K$. There is a pseudo-versal functor to groupoids over the functor of $K$-points
for family of functors with properties (0-3) which satisfies the unique path lifting property, namely
%
%There is a functor $\pi_1:\VV\lra \Groupoids$ to groupoids satisfying (0-3) such that for each each functor $\pi:\VV\lra Groupoids$ 
%satisfying (0-3) there is a field automorphism $\sigma:K\lra K$ and
%a natural transformation $\varepsilon: \pi_1 \Lra \sigma(\pi)$ such that
%$\varepsilon: \Points \pi_1^{top}(X(K)) = \Points \sigma(\pi)(X(K))$, 
%and, moreover, $\pi_1$ satisfies the unique path lifting property, namely
 \begin{itemize}
   \item[$(3^!)$] for $X\xra f Y$ \'etale, the map $\pi(X)\xra {f_*}\pi(Y)$ of groupoids
                has the {\em unique} path lifting property of topological covering maps, namely
          \begin{itemize}
          \item[]   for $y=f(x),x\in X(K)$, for every path $\gamma\in \pi(X)$ starting at $x$,
                there is a {\em unique} path $\tilde\gamma\in \pi(Y)$ starting at $y$
                such that $f_*(\tilde\gamma)=\gamma$.
          \end{itemize}
\end{itemize}
\end{enonce*}


\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage[matrix,arrow,curve]{xy}
%\textwidth=16.5 cm
%\textheight=21.5 cm
%\hoffset=-1cm
%\voffset=-2cm

\def\Het{H_{et}}
\def\Ql{{\Bbb Q}_l}

\def\Lwlw{L_{\omega_1\omega}}
\def\Hsing{H_{\text{sing}}}
\def\Q{{\Bbb Q}}
\def\Qbar{\bar{\Bbb Q}}



\def\lra{\longrightarrow}
\def\Lra{\Longrightarrow}
\def\xra{\xrightarrow}
\def\xla{\xleftarrow}


\def\C{\Bbb C}
\def\R{\Bbb R}
\def\Z{\Bbb Z}


\def\Qvect{\Q\text{-Vect}}
\def\QVect{\Q\text{-Vect}}
\def\QHodge{\Q\text{-Hodge}}


\def\ZlVect{\Q_l\text{-Vect}}
\def\QpVect{\Q_p\text{-Vect}}

\def\id{id}

\def\Var{\mathcal Var}
\def\Mot{\mathcal Mot}
\def\V{\mathcal V}
\def\VV{\mathcal V}
\def\WW{\mathcal WW}
\def\W{\mathcal W}
\def\CC{\mathcal C}

\def\Het{H_{et}}

\def\tensor{\otimes}

\def\Gal{Gal}
\def\Aut{Aut}

\def\ex{\mathrm{ex}}
\def\Ker{\mathrm{Ker}\,}


\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{example}{Example}[section]
\newtheorem{question}{Question}[section]


\begin{document}
\date{}
\author{notes by Misha Gavrilovich
\thanks{I think  A.Luzgarev, V.Sosnilo, and particularly M.Bays
for many discussions. The only proof in this note was 
workout together with Martin Bays.}
}

\title{Standard conjectures on categoricity of cohomologies}
\maketitle

\begin{abstract}
An analytic or topological construction of a functor
in algebraic geometry involves a choice of a topology
on the ground field or its extension, in other words,
a choice of an embedding to the (topological) field of 
complex numbers. Thus, from a purely algebraic point of view,
%such a construction defines _a family_ of functors,
such a construction defines {\em a family} of functors,
which are not equivalent yet, by construction, 
possess the same algebraic properties. 

We formulate a conjecture that each Q-form of the \'etale
%cohomology functor X |---> H_et(X,Q_l) come from a compasison
cohomology functor $X\mapsto  \Het(X,\Ql)$ come from a compasison
isomorphism between singular cohomology and \'etale cohomology. 
 We prove a very limited case of this conjecture and observe
 the conjecture appears to be related to the 
 Grothenidick Standard Conjectures  and 
% the conjectures on the motitivic Galois group   Aut(H_sing(-,Q)) 
 the conjectures on the motitivic Galois group   $Aut(\Hsing(-,\Q))$ 
and its l-adic representations.
\end{abstract}


%## Introduction
\section{Introduction}


Often a construction of a functor involves arbitrary choices and 
%thereby produces _a family_ of functors, which are different yet, 
thereby produces {\em a family} of functors, which are different yet,
by the very nature of their construction, 
possess the same properties as algebraic structures; such are analytic or
topological constructions of functors, from the category of algebraic varieties
which involve a choice of topology or a choice of an embedding to the field of
complex numbers. It makes sense to ask whether (being a member of) 
%such a _family_ (a heresy!) can be characterised purely in terms of 
such a {\em family} (a heresy!) can be characterised purely in terms of
"algebraic" properties. This question is analogous to the question of 
categoricity in model theory.



In this short letter we discuss two examples, 
the family of cohomology theories 
%  H_top ( X(K_\tau), Q )  
  $\Hsing(X(K_\tau),\Q)$
and the family of group homomorphisms 
% \tau^{-1}  exp \circ \tau : K^+ --> K^*
$ \tau^{-1}  exp \circ \tau : K^+ \lra K^*$
%where \tau varies among isomorphisms \tau:K--->C.
where $\tau$ varies among isomorphisms $\tau:K\lra C$.

The latter is known in model theory under the name
of pseudo-exponentiation and is well-understood [BaysKirby,Zilber].
In \S1 we explain the categoricity theorem of [BaysKirby,Zilber]
from our point of view. 

In \S2 formulate a conjecture that each Q-form of the \'etale
%cohomology functor X |---> H_et(X,Q_l) come from a compasison
cohomology functor $X\mapsto  \Het(X,\Ql)$ come from a compasison
isomorphism between singular cohomology and \'etale cohomology. 
 We prove a very limited case of this conjecture and observe
 the conjecture appears to be related to the 
 Grothenidick Standard Conjectures  and 
% the conjectures on the motitivic Galois group   Aut(H_sing(-,Q)) 
 the conjectures on the motitivic Galois group   $Aut(\Hsing(-,\Q))$ 
and its l-adic representations.


In \S4 we present several vague questions and speculations. 
Namely, we ask whether (being a member of)
either of the families
% H_top ( X(K_\tau), Q ) : Var/K ---> Q-vect
% H ( X(K_tau),C) : Var/K ---> Q-Hodge
 $$\Hsing( X(K_\tau), \Q ) : \Var/K \lra \Qvect,$$  
 $$\Hsing( X(K_\tau), \C ) : \Var/K \lra \QHodge$$
can be characterised purely in terms of "algebraic"
properties of functors and/or cohomology theories. 
We add some further speculations on how 
such a characterisation could be
related to known conjectures.


%## Categoricity for Pseudo-exponentiation.
\section{Model theory: categoricity for pseudo-exponentiation}

Let us describe the complex exponential function exp:C-->C 
from this point of view.

There is no "canonical" group homomorphism from the additive group
%of an algebraically closed field K to its multiplicative group. However, 
of an algebraically closed field $K$ to its multiplicative group. However,
%it is easy to construct one if the field K admits 
it is easy to construct one if the field $K$ admits
a locally compact Hausdorff locally connected topology 
compatible with the field operations, or, equivalently, 
is isomorphic to the field of complex numbers. 
Can one characterise (members of) the family of homomorphisms
%ex: K^+ --> K^* that could be obtained this way, or, equivalently, 
$ex: K^+ \lra K^*$ that could be obtained this way, or, equivalently,
%\sigma^{-1} exp\circ \sigma : C^+ --> C^* for \sigma : C-->C 
$\sigma^{-1} exp\circ \sigma : \C^+ \lra \C^*$ for $\sigma : \C\lra\C $
a (possibly discontinuous, non-measurable etc)
%field automorphism of C ?
field automorphism of $\C$ ? 

A categoricity theorem [BaysKirby,Zilber] of Boris Zilber provides such a characterisation
assuming some well-known conjectures. To provide such a characterisation,
we first need to define what it means to be a member of the family, 
a notion of sameness or isomorphism. As is standard in model theory,
here we consider first-order structures in the language with symbols
% ( K, +:KxK-->K, *:KxK-->K, ex:K-->K )
$$ ( K, +:K\times K\lra K, *:K\times K\lra K, ex:K\lra K ) $$
and say two such structures are isomorphic iff there is a bijection 
%\sigma: K-->K' which commutes with the functions listed explicitly.
$\sigma: K\lra K'$ which commutes with the functions listed explicitly.
"Algebraic" properties are taken to be properties expressible
%in this language in an Lw1w(Q)-logic. In fact, 
in this language in an $\Lwlw(\Q)$-logic. In fact,
there is an explicit description of the algebraic, "discrete"
%properties of a group homomorphism ex: K^+ --> K^*
properties of a group homomorphism $ex: K^+ \lra  K^*$
which includes Schanuel conjecture from the transcendental number theory.

The proof relies on the transitivity of Galois action 
on the roots of unity and Schanuel conjecture,  
which are related to Mumford-Tate conjecture 
and Grothendieck conjecture on periods
formulated in terms of motivic Galois group
(i.e. the group of automorphisms of a cohomology theory
of algebraic varieties). Note that the latter 
conjectures are used in our discussion of categoricity
of singular cohomology of complex algebraic varieties
(There is a generalisation of  Schanuel conjecture 
in the functorial language [Andre04, ch. 23], also 
cf. Grothendieck conjecture on periods [Andre08, 4.1.1].)

Note that Zilber also proves an unconditional result,
namely that for every two group homomorphisms 
%ex': K^+ --> K^*, ex'' : K^+ --> K^*
$ex': K^+\lra K^*, ex'':  K^+\lra K^*$
satisfying certain properties, there is a field 
%automorphisms sigma:K-->K such that
automorphisms $\sigma:K\lra K$ such that 
%ex' = \sigma^{-1} ex'' \circ \sigma .
$ex' = \sigma^{-1} ex'' \circ \sigma $



%## Categoricity for the singular  cohomology of complex algebraic varieties
\section{Categoricity for the comparison isomorphism between  singular 
and \'etale cohomology of complex algebraic varieties}


\subsection{Comparison isomorphism as a $\Q$-form of \'etale cohomology} 

Let $\QVect$ and $\ZlVect$ denote the category of 
finite dimensional vector spaces over $\Q$, resp. over $\Q_l$.


A {\em $\Q$-form} of a functor $H_l: \V \lra \ZlVect$ is a pair $(H,\tau)$ 
consisting of a functor $H:  \V \lra \QVect$ and 
a {\em comparison} isomorphism 
$$\tau: H(-)\tensor \Ql  \Lra H_l $$ 
For any two $\Q$-forms $(H',\tau'),(H'',\tau'')$ of $H_l$, 
a natural transformation $\alpha: H'\Longrightarrow H''$ 
induces a natural transformation 
$$\alpha\tensor\Ql:  H'(-)\tensor \Ql \Lra  H''(-)\tensor \Ql$$
and hence an automorphism $$\tau'^{-1} (\alpha\tensor\Ql) \tau'': H_l\Lra H_l.$$ 
We say two $\Q$-forms $(H',\tau')$ and $(H'',\tau'')$ of $H_l$ are {\em equivalent}
iff this automorphism is identity: $\tau'^{-1} (\alpha\tensor\Ql) \tau''=\id$.
$$ H'(-)\tensor \Ql \xra{\tau'} H_l(-) \xleftarrow{\tau''}  H''(-)\tensor \Ql$$

One may think about $\Q$-forms as subfunctors $H(-)\subseteq H_l(-)$ such that
the $\Q$-vector space $H(-)$ is dense and of the same dimension.


Let $\Het^i(X,\Z/N\Z)$, $N$ an integer,  denote the \'etale cohomology group [Katz,p.21]; 
they are well-defined for an {\em arbitrary} scheme $X$. This is a functor 
from the category of (arbitrary) schemes. 
Let $\Hsing^i( X(\C), \Q )$ denote the singular cohomology group of the space $X(\C)$
of complex points of the scheme $X$ of finite  type over $\C$ 
equipped with the complex topology.




There is a canonical comparison isomorphism
$$\Hsing^i( X(\C), \Q ) \tensor \Q_l = \Het^i ( X, \Q_l ), \ \ 0\leq i\leq 2\dim X$$
for $X$ a separated scheme over $\C$ of finite type (cf. [Katz, SGA4, XVI, 4.1]). 

This shows that the singular cohomology theory with $\Q$-coefficients 
is a $\Q$-form of the \'etale cohomology theory.


Etale cohomology is a functor from the category of (arbitrary) schemes and therefore
are functorial wrt field automorphisms $\sigma : K\lra K$, eqv. $\sigma : Spec K \lra Spec K$.

This lets to define the action of $Aut(K)$ on the $\Q$-forms $(H,\tau)$ of $\Het_{|\Var/K}$
by 
$$\sigma(H,\tau) =  (H\circ \sigma^{-1}, \Het(\sigma,\Ql) \circ \tau ).$$ 
In terms of subfunctors, it is $\Het(\sigma,\Ql)H\circ \sigma^{-1} \subseteq \Het(X,\Ql)$. 



Further, it is convenient to think of a comparison isomorphism
as a subfunctor H of $\Het$ such that 
each $\Q$-vector space $H^i(X) \subseteq \Het^i(X)$ is dense and 
is of the same dimension.





The discussion above may be summed up as follows.
A choice of a connected locally connected locally compact topology $\tau$
on an algebraically closed field $K$ with continuous field operations,
gives rise to a $\Q$-form of \'etale cohomology
$$\Hsing^i( X(K_{\tau}), \Q ) \tensor \Q_l  \xra[\tau_*]{=} \Het^i ( X, \Q_l ),\ \  0\leq i\leq 2\dim X,$$ 
for  $X$ a separated scheme over $K$ of finite type.

(A field with such a topology is necessarily isomorphic to either $\C$ or $\R$, and there is a unique continuous isomorphism, 
up to complex conjugation.)




\subsection{A conjecture on categoricity of the comparison isomorphism between the singular and \'etale cohomology}



Let $\Var/K$ denote the category of smooth projective separated 
schemes of finite type over a field $K$, 
and let $Mot/K$ be the category of pure motives as defined in [Kleiman];
Standard Conjectures are required in order
for the category of pure motives to have good properties. 
%there is a canonical functor Var/K --> Mot/K. 
There is a canonical functor $\Var/K \lra Mot/K$.



We conjecture that each $\Q$-form of the \'etale cohomology
is associated to the singular cohomology theory via 
a comparison isomorphism $\tau_*$.


\begin{enonce*}{Conjecture}[$E_\C(\Ql)$] 
%Conjecture(Categoricity for comparison isomorphisms): 
% Let H: Var/K ---> Q-vect be a functor which admits an isomorphism



  Let $K$ be a field isomorphic to the field of complex numbers. 
  Assume that both $\Hsing$ and $\Het$ factor via the category $\Mot/K$ 
  of pure motives over K. 
 
  Let $H: \Mot/K \lra \QVect$ be a functor.

 Any isomorphism of functors $\Mot/K\lra \ZlVect$ 
%      H( X, Q ) \tensor Q_l == H_et ( X, Q_l )
  $$H(-) \tensor_\Q \Ql \xra{\tau} \Het(-,\Ql)$$
comes from topology,
i.e. there is a locally compact locally connected topology 
$\tau_H$ on the field $K$ 
and a natural transformation 
  %  H ==> H_top ( X(K)_top, Q)
 $$ H \Lra \Hsing( X(K_{\tau_H}),\Q)$$
%which induces identity on H_et ( X, Q_l ) via the isomorphisms
which induces identity on $\Het ( X, \Ql )$ via the isomorphisms
%    H( X ) \tensor Q_l == H_et ( X, Q_l ) == H_top( X, Q ) \tensor Q_l .
$$ H(-)\tensor \Ql \xra{\tau'} H(-) \xleftarrow{\tau''}  H''(-)\tensor \Ql$$  
\end{enonce*}

One may avoid mentioning topology by claiming that all the $\Q$-forms are related by
$Aut(K)$-action.



\begin{enonce*}{Conjecture}[$E(\Ql)$] 
%Conjecture(Categoricity for comparison isomorphisms): 
% Let H: Var/K ---> Q-vect be a functor which admits an isomorphism

  Let $K$ be an algebraically closed field. 
  Assume that both $\Hsing$ and $\Het$ factor via the category $\Mot/K$ 
  of pure motives over K. 

 Then  $Aut(K)$ acts transitively on the $\Q$-forms of $\Het_{|\Mot/K}$ considered on the category $\Mot/K$ of pure motives.
 That is, for two $\Q$-forms $(H',\tau'),(H'',\tau'')$ of $\Het_{|\Mot/K}$ considered on the category $\Mot/K$ of pure motives
there is a field automorphisms $\sigma\in Aut(K)$ such that
$(H'',\tau'')$ is equivalent to $(H'\circ \sigma^{-1}, \Het(\sigma,\Ql)\circ\tau')$.
\end{enonce*}


\subsection{A conjecture on categoricity of the comparison isomorphism between the singular and \'etale cohomology}

Consider \'etale cohomology on smaller subcategories to obtain more manageable conjectures.

 Below we do so and consider $H$ and $\Het$ restricted
to a subcategory generated by a single Abelian variety.


Let $\V,\W\subseteq \Mot/K$ be subcategories,
and let $k$ be a field such that $Aut(K/k)$ acts on $\V$. 


\begin{enonce*}{Conjecture}[$E(\Ql, \VV)$] 
$Aut(K/k)$ has finitely many orbits on the $\Q$-forms of the \'etale cohomology theory 
$\Het_{|\V}:\V\lra\ZlVect$ 
restricted to the subcategory $\V$.
\end{enonce*}

\begin{enonce*}{Conjecture}[$E_{weak}(\Ql, \VV)$]   
$Aut(\Het_{|\V})$ has finitely many orbits on the $\Q$-forms of the \'etale cohomology theory 
$\Het_{|\V}:\V\lra\ZlVect$
restricted to the subcategory $\V$. 
\end{enonce*}


\begin{enonce*}{Conjecture}[$E(\Ql, \W\subset \VV)$]

Let $H:\W\lra \QVect$ be a $\Q$-form of $\Het_{|\W}:\W\lra\ZlVect$.
Then $Aut(K/k)$ has finitely many orbits on the $\Q$-forms $H':\V\lra \QVect$ of the \'etale cohomology theory 
$\Het_{|\V}:\V\lra\ZlVect$ such that  $H'_{|\W} = H_{|\W}$.
\end{enonce*}

\begin{enonce*}{Conjecture}[$E_{weak}(\Ql, \W\subset \VV)$]   
Let $H:\W\lra \QVect$ be a $\Q$-form of $\Het_{|\W}:\W\lra\ZlVect$.
Then $Aut(K)$ has finitely many orbits on the $\Q$-forms $H':\V\lra \QVect$ of the \'etale cohomology theory 
$\Het_{|\V}:\V\lra\ZlVect$ such that  $H'_{|\W} = H_{|\W}$.
\end{enonce*}


\subsection{An example: an Abelian variety.}

Let $k$ be a number field and let $A$ be an Abelian variety defined over $k$. 
$\left<A\right>$ denote the least Tannakian subcategory of $\Mot/k$
containing the motive of $A$. 
Finally, let $\left<A\right>\tensor\Qbar$ denote the subcategory of $\Mot/\Qbar$
corresponding to $\left<A\right>$.



\begin{example} $E_{weak}(\Ql, \left<A\right>\tensor\Qbar)$ holds
for $A$ an Abelian variety defined over a number field $k$
such that its Mumford-Tate group is $GSp_{2g}$ where $\dim A=g$. 
\end{example}

%Remark/Expectation:
%    If H is a Q-form of H_et, then \Aut(H) is a pro-algebraic group
%    over Q, and \Aut(H) \tensor_\Q \Q_l ~= \Aut(H_et).
%
%    Perhaps some sort of converse also holds?
%
%Example:
%    Conjecture A holds for $A$ an Abelian variety, $\dim A=g$, such that the
%    Mumford-Tate group is $\Symp_{2g}$.
%Proof (sketch):
{\bf Proof} (sketch).
The Weil pairing corresponds to a motive, 
    and by functoriality we have that, 
    for a $\Q$-form $H$ of $\Het |_{<A>}$,
    the non-degenerate Weil pairing
%        $\omega : (\Het^1(A))^2 -> \Het^0(A) ~= \Q_l$
$$\omega : (\Het^1(A))^2 \lra \Het^0(A) = \Q_l$$
restricts to a pairing
%        $\omega : (H^1(A)^2 \lra H^0(A) \approx \Q$,
 $$\omega : (H^1(A))^2 \lra H^0(A) \approx \Q,$$
which is easily seen to be non-degenerate.

    Now let $H_i$ be a $\Q$-form for $i=1,2$.

    Let $(x_i^1,...x_i^g,y_i^1,...,y_i^g)$ be a symplectic basis for $H_i^1(A)$.
    Then each is also a symplectic basis for $\Het^1(A)$,
    and so some $\sigma \in GSp(\Het^1(A),\omega)$ maps $H_1^1(A)$ to $H_2^1(A)$.
    The assumption on the Mumford-Tate group precisely means that such a
    $\sigma$ extends to $\sigma \in \Aut(\Het |_{\left<A\tensor\Qbar\right>})$,
    and it follows from the fact that the cohomology of an Abelian variety is
    generated by $H^1$ that $\sigma(H_1) = H_2$
%    (TODO: details).


The proof above probably generalises to the following.

\begin{enonce*}{Conjecture}[a generalisation of the example]
%Conjecture (a generalisation of the example):
%   Assume the Mumford-Tate group G=Aut(H_et|<A>)  has the following property:
%if V_1 and V_2 are Q-vector subspaces of H_et(A,Q_l)
%        such that GL(V_i) /\ G(Q_l) is dense in G(Q_l) for i=1,2,
%        then there is a g in G(Q_l) such that gV_1=V_2 (setwise).
%
%    Then A  holds for H_et|<A>.

 Let $A$ be a motive of a smooth projective variety defined over a number field $k$. 
 Assume the Mumford-Tate group $G=Aut(\Het_{|\left<A\right>\tensor\Qbar})$
has the following property: 
 \begin{itemize}
 \item[] if $V_1$ and $V_2$ are $\Q$-vector subspaces of $\Het(A\tensor\Qbar,\Q_l)$
         which are both dense and of the same dimension and 
         such that 
	 \begin{itemize}\item[]$GL(V_i) \cap G(\Q_l)$ is dense in $G(\Q_l)$ for i=1,2,
	 \end{itemize}
        then there is a $g \in G(\Q_l)$ such that $gV_1=V_2$ (setwise).
 \end{itemize}
     Then $E_{weak}(\Q_l,\left<A\right>\tensor\Qbar)$  holds.
 \end{enonce*}


\subsection{Standard Conjectures and motivic Galois group}

Now we try to place the conjectures above in the context
of Standard Conjectures and the conjectures
on the motivic Galois group [Keiman, Serre].
Our exposition follows [Serre,\S1,\S3]


Let $k$ be a field of characteristic 0 which embeds into the field $\C$ of complex numbers; pick an embedding $\sigma:k\lra\C$. 

Assume Standard Conjectures and Hodge conjecture [Grothendieck, Kleiman]. 
Let $\Mot_k$ denote the category of pure motives over $k$ 
defined with the help of numerical equivalence of algebraic cycles
(or the homological equivalence, which should be the same 
by Standard Conjectures). $\Mot_k$ is a semi-simple category. 


Let $E\in Ob \Mot_k$ be a motive; let $\left<E\right>$ denote
the least Tannakian subcategory of $\Mot_k$ containing $E$.

A choice of embedding $\sigma:k\lra\C$ defines 
an exact {\em fibre functor} 
$\Mot\lra\QVect$ corresponding to {\em the Betti realisation} 
$${\Hsing}_\sigma: \Mot\lra \QVect,\ \ \ E\mapsto \Hsing(E_\sigma(\C),\Q).$$ 

The scheme of automorphisms 
$MGal_{k,\sigma}=\Aut^\tensor({\Hsing}_\sigma:\Mot\lra\QVect)$ 
of the functor preserving the tensor product
is called {\em motivic Galois group of $k$}.
It is a linear proalgebraic group defined over $\Q$. 
Its category of $\Q$-linear representations is equivalent to $\Mot_k$. 
The group depends on the choice of $\sigma$. 

The motitivic Galois group of a motive $E$ is 
$\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)$. 


Serre conjectures
\begin{enonce*}{Conjecture}[2.1?] The group $\Aut^\tensor({\Hsing}_\sigma:\Mot\lra\QVect)$ is proreductive, i.e. a limit of liner reductive $\Q$-groups.
\end{enonce*}

Let $1$ denote the trivial morphism of rank 1, i.e. the cohomology 
of the point $Spec\, k$. 

\begin{enonce*}{Conjecture}[3.1?] The group $\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)$ is the subgroup of $GL(\Hsing_sigma(E)$ 
preserving the tensors corresponding to morphisms
 $1\lra E^{\tensor r} \tensor E^{\vee\tensor s}$, $r,s\geq 0$. 
\end{enonce*}

It is also conjectured that this group is reductive. 
Via the comparison isomorphism of \'etale and singular cohomology, 
$$\Hsing( E(\C), \Q ) \tensor \Q_l = \Het ( E\tensor\Qbar, \Q_l )$$,
the $\Ql$-points of $\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql)$ act on the \'etale cohomology $\Het ( E\tensor\Qbar, \Q_l )$.
On the other hand, the Galois group $Gal(\Qbar/k)$ acts
on $\Qbar$ and therefore on $E\tensor\Qbar$. 
By functoriality, the Galois group acts by automorphisms of the functor of \'etale cohomology. 
 Hence, this gives rise to {\em $\ell$-adic representation associated to $E$}
  $$\rho_{k,l}: Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql).$$




\begin{enonce*}{Conjecture}[3.2?] Let $k$ be a number field. 
The group $\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql)$ 
is dense in the Zariski topology in the image 
of the $\ell$-adic representation associated with $E$. 
\end{enonce*}


\begin{enonce*}{Conjecture}[9.1?] Let $k$ be a number field. 
The image $$
Im(\rho_{k,l}: 
Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql))
$$ 
is open in $\Aut^\tensor({\Hsing}_\sigma:\left<E\right>\lra\QVect)(\Ql)$.
\end{enonce*}


\begin{enonce*}{Conjecture}[9.3?] Let $k$ be a number field. 
$ \Het ( E\tensor\Qbar, \Q_l )$ is semi-simple as a $Gal(\Qbar/k)$-module. 
\end{enonce*}

We suggest that the  conjectures [2.1,3.1,3.2,9.1]
may be interpreted as saying there are only finitely
many $\Q$-forms of the \'etale cohomology 
$\Het(-,\Ql): \left<E\right>\tensor_k\Qbar\lra \QVect$,
up to Galois action. 

%%The following conjecture is probably easy to prove. 
%%
%%\begin{enonce*}{Conjecture}[$E(\Ql, \left<E\right>\tensor_k\Qbar)$]. 
%%Let $k$ be a number field and let $E$ be a motive over $k$. 
%%Assume that the action of the motivic Galois group $G=MGal(E)$ of $E$ 
%%on \'etale cohomology satisfies the following condition. 
%% \begin{itemize}
%%  \item[] if $V_1$ and $V_2$ are $\Q$-vector subspaces of $\Het(E\tensor\Qbar,\Q_l)$
%%   which are both dense and of the same dimension and
%%       such that
%%        \begin{itemize}\item[]$GL(V_i) \cap G(\Q_l)$ is dense in $G(\Q_l)$ for i=1,2,
%%            \end{itemize}
%%      then there is a $g \in G(\Q_l)$ such that $gV_1=V_2$ (setwise).
%%  \end{itemize}
%%
%%
%%Then the conjectures [2.1,3.1,3.2,9.1] above imply that 
%%there are finitely many $\Q$-forms of the \'etale cohomology functor
%%$\Het(-,\Ql): \left<E\right>\tensor_k\Qbar\lra \QVect$. 
%%\end{enonce*}
%%
 There are similar conjectures for finite adels instead of $\Q_l$,
  cf. [Serre, 11.4?(ii), 11.5?], cf. also [Serre, 10.2?, 10.6?].
We suggest they may interrupted as
$E({\Bbb A}^f, \left<E\right>\tensor_k\Qbar)$ 
where ${\Bbb A}^f$ is the ring of finite adeles.

\section{Speculations and remarks}


Standard conjectures claim there are algebraic cycles corresponding to various cohomological constructions. Model-theoretically it should mean
that something is definable in ACF and it is natural to expect
that such properties be useful in a proof of categoricity, i.e.~in the characterisation of the $\Q$-forms of \'etale cohomology theory. 

We wish to specifically point out the conjectures and properties 
involving smooth hyperplane sections, namely
{\em weak and strong Lefschetz theorems} and {\em Lefschetz Standard Conjecture}, cf.~[Kleiman,p.11,p.14]. 
Weak  Lefschetz theorem describes 
part of the cohomology ring of a smooth hyperplane section of a variety. Perhaps such a description can be useful in showing 
that a $\Q$-form extends uniquely to $\Mot/K$ from the subcategory $\Mot/\Qbar$. An analogue of the weak Lefschetz theorem for the fundamental group was used in a similar way in [Gavrilovich].



\begin{question} Find a characterisation 
of the following families of functors:
% H_top ( X(K_\tau), Q ) : Var/K ---> Q-vect
% H ( X(K_tau),C) : Var/K ---> Q-Hodge
 $$\Hsing( X(K_\tau), \Q ) : \Var/K \lra \Qvect,$$
  $$\Hsing( X(K_\tau), \C ) : \Var/K \lra \QHodge$$
where $\tau$ varies though isomorphisms of $K$ to $\C$,
or, almost equivalently, though locally compact locally 
connected topologies on $K$.
\end{question}

Note that Zilber {\em uncoditionally} constructs a pseudo-exponential map $ex:\C^+\lra\C^*$ which satisfies the Schanuel conjecture. Of course, this map is not continious (not even  measurable). 
 Hence we ask:
\begin{question} Construct a pseudo-singular cohomology theory 
which satisfies an analogue of the Schanuel conjecture and some other conjectures.
\end{question}


\subsection{Model theoretic conjectures}

Define model theoretic structures corresponding to the cohomology 
theories.


\begin{enonce*}{Conjecture}
   The field is purely embedded into the structures
   corresponding to functors
\begin{itemize}
   \item[(i)]     $\Hsing:Var/Qbar \lra \QVect$
   \item[(ii)]    $\Hsing(-,Q):Var/C ---> \QVect$
   \item[(iii)]   $\Hsing(-,\C):Var/C ---> \QHodge$
\end{itemize}
Moreover, the structure (ii) is an elementary extension of (i) and
the cohomology ring $\Hsing(V,Q)$ is definable for every variety over C.
\end{enonce*}


Several of the Standard Conjectures [Kleiman, \S4,p.11/9] claim that 
certain cohomological cycles (construction) correspond to algebraic 
cycles. This feels related to many of the conjectures above, 
in particular to the purity conjectures.


\subsection{TODO:}
\begin{itemize}
\item[1.] Define a model-theoretic structure and language corresponding
 to the notion of a Weil cohomology theory, and formulate 
 a categoricity conjecture hopefully related to the Standard Conjectures ([Grothendieck, Kleiman])
 and conjectures on the motivic Galois Group and related Galois representations [Serre].

 \item[2.] Do the same in the language of functors, namely:
  \begin{itemize}
  \item[2.1]. Consider the family of cohomology theories on 
  $\Var/$K coming  from a choice of isomorphism $K\approx\C$.
  \item[2.2.] Define a notion of isomorphism of these/such cohomology theories,
    and  what it means to a "purely algebraic" property of such a theory.
   \item[3.3.] Find a characterisation of that family up to that notion of isomorphism
   by such properties. Or rather, see it is equivalent to a number of well-known conjectures
   such as the Standard Conjectures etc.
\end{itemize}
\end{itemize}




\begin{thebibliography}{10}

\bibitem[Andre04]{Andre04} Andre, Yves, Une introduction aux motifs (motifs purs, motifs mixtes, periodes).                     
 Panoramas et Syntheses 17. Societe Mathematique de France, Paris, 2004.                                        


\bibitem[Andre08]{Andre08} 
Galois theory, motives and transcendental numbers                                 http://arxiv.org/abs/0805.2569                                                                                 
\bibitem[BaysDPhil]{BaysDPhil}
Categoricity results for exponential maps of 1-dimensional algebraic groups & Schanuel Conjectures for Powers and the CIT.
Oxford, DPhil thesis. 
\url{https://ivv5hpp.uni-muenster.de/u/baysm/dist/thesis/}

\bibitem[BZ]{BZ}
Martin Bays, Boris Zilber.
Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic.
 Bull London Math Soc (2011) 43 (4): 689-702.  DOI: \url{https://doi.org/10.1112/blms/bdq131}
\url{https://arxiv.org/abs/0704.3561}


\bibitem[BK]{BK} 
Martin Bays, Jonathan Kirby. 
Excellence and uncountable categoricity of Zilber's exponential fields.
http://arxiv.org/abs/1305.0493

\bibitem[DawHarris]{DawHarris}
Christopher Daw, Adam Harris. 
Categoricity of modular and Shimura curves.
Journal of the Institute of Mathematics of Jussieu (2015) pp. 1-27
\url{https://doi.org/10.1017/S1474748015000365}
\url{https://arxiv.org/abs/1304.4797}

\bibitem[Zilber]{Zilber} B.Zilber,  Pseudo-exponentiation on algebraically closed fields of characteristic zero Annals of Pure and Applied Logic, Vol 132 (2004) 1, pp 67-95. https://people.maths.ox.ac.uk/zilber/expf.html


\bibitem[GavrDPhil][GavrDPhil]
Misha Gavrilovich.
 Model theory of the universal covering spaces of complex algebraic varieties.
Oxford, DPhil thesis.
\url{http://mishap.sdf.org/misha-thesis.pdf}

\bibitem[GavrK][GavrK]
Misha Gavrilovich.
A remark on transitivity of Galois action on the set of uniquely divisible Abelian extensions in $Ext^ 1 ( E (\mathbb Q), \Lambda )$.
K-Theory (2008) 38:135–152 \url{DOI https//doi.org/10.1007/s10977-007-9015-0}

\bibitem[Grothendieck]{Grothendieck}
Standard Conjectures on Algebraic Cycles, 
   http://mishap.sdf.org/temp/Grothendieck\_Standard\_Conjectures\_on\_Algebraic\_Cycles.pdf

\bibitem[HarrisDPhil]{HarrisDPhil}
Adam Harris.
 Categoricity and covering spaces.
 Oxford, DPhil. 



\bibitem[Serre]{Serre} Properties conjecturales des groupes de Galois motiviques et des representations l-adiques.              
 http://mishap.sdf.org/temp/Serre-Motivic-Galois.pdf                                                           
                                                                      
								      
								      
								      \bibitem[Kleiman]{Kleiman} Standard Conjectures.  http://mishap.sdf.org/temp/Kleiman\_Standard\_Conjectures.pdf                                                   
                                                                                                                
 \bibitem[Katz]{Katz} Review of l-adic cohomology                        http://mishap.sdf.org/temp/Katz\_Review\_of\_l-adic\_cohomology.pdf                                               
 
  (in Uwe Jannsen, Steven L. Kleiman. Motives (Proceedings of Symposia in Pure Mathematics) (Part 1). 
  American+Mathematical Society (1994) )                                                                                          

\end{thebibliography}


\end{document}

It is conjectured that the image of $\Gal(\bar{k_0}/k_0)$ in the motivic 
Galois group $MT(X)$ is open, and it is furthermore conjectured that 
$\Aut(\Het |_{<X>}) = MT(\Q_l)$; see [Serre, 2.1?, 9.1?].

These conjectures and  $E_{weak}(\Q_l,\left<X\tensor_{k_0}\Qbar\right>)$ imply $E(\Q_l,\left<X\tensor_{k_0}\Qbar\right>)$.
In other word, 


%Indeed, if also Conjecture $E_{weak}(\Q_l,\left<X\tensor_{k_0}\Qbar\right>)$ holds for $\Het |_{<X>}$, 
%it follows that up to the action of $\Gal(k_0)$, there are only finitely many subfunctors 
%$H$ of  $\Het |_{<X>}$ such that each $H^i(Y) <= \Het^i(Y)$ is dense and has the same
%dimension. It is important here that the source category is a category of
%(pure) motives.


Remark:
 There are similar conjectures for finite adels instead of Q_l, 
 cf. [Serre, 11.4?(ii), 11.5?], cf. also [Serre, 10.2?, 10.6?]




















%Conjecture(Categoricity for comparison isomorphisms): 
% Let H: Var/K ---> Q-vect be a functor which admits an isomorphism



  Let $K$ be a field isomorphic to the field of complex numbers. 
  Assume that both $\Hsing$ and $\Het$ factor via the category $\Mot/K$ 
  of pure motives over K. 
 
  Let $H: \Mot/K \lra \QVect$ be a functor.

 Any isomorphism of functors $\Mot/K\lra \ZlVect$ 
%      H( X, Q ) \tensor Q_l == H_et ( X, Q_l )
  $$H(-) \tensor_\Q \Ql \xra{\tau} \Het(-,\Ql)$$
comes from topology,
i.e. there is a locally compact locally connected topology 
$\tau_H$ on the field $K$ 
and a natural transformation 
  %  H ==> H_top ( X(K)_top, Q)
 $$ H \Lra \Hsing( X(K_{\tau_H}),\Q)$$
%which induces identity on H_et ( X, Q_l ) via the isomorphisms
which induces identity on $\Het ( X, \Ql )$ via the isomorphisms
%    H( X ) \tensor Q_l == H_et ( X, Q_l ) == H_top( X, Q ) \tensor Q_l .
$$ H(-)\tensor \Ql \xra{\tau'} H(-) \xleftarrow{\tau''}  H''(-)\tensor \Ql$$  
\end{enonce*}





\begin{enonce*}{Conjecture}[$E_K$] 
\end{conjecure}
Let H_et(-,Q_l) denote the functor of \'etale cohomology
which sends a variety X into the Q_l-vector space 
H_et^0 (X,Q_l) \oplus ... \oplus H_et^{2 \dim X}(X,Q_l). 




Here we formulate the conjecture that it is categorical to describe 
the singular  cohomology of complex algebraic varieties 
as a Weil cohomology theory with $\Q$-coefficents which admits
a comparison isomorphism with the \'etale cohomology.




# A naive categoricity for the singular  cohomology of complex algebraic varieties
\subsection{A naive categoricity for the singular  cohomology of complex algebraic varieties}

A naive conjecture is that the following description is categorical:

%Conjecture (a naive vague conjecture on Categoricity of the singular  cohomology of complex algebraic varieties):
\begin{enonce*}{Conjecture}[(a naive vague conjecture on Categoricity of the singular  cohomology of complex algebraic varieties]
Fix an algebraically closed field K. 
  The following description is categorical: 
  a Weil cohomology theory on Var/K with Q-coefficients which factors 
  via the category of pure motives Mot/K.
\end{enonce*}

In other words, a Weil cohomology theory with Q-coefficients which factors 
via the category of pure motives, necessarily has form 
%H_top ( X_sigma(K), Q ) for some field isomorphism sigma:K-->C.  
$\Hsing(X_\sigma(\C), \Q$ for some field isomorphism $\sigma:K\lra \C$. 

Note, however, that the conjecture is vague, as the notion of 
"isomorphism" ("be of form") of cohomology theories is not specified.
It is also quite likely that we need to require more properties. 





%## Categoricity for the comparison isomorphism between \'etale cohomology and
%## the singular  cohomology of complex algebraic varieties
\section{Categoricity for the comparison isomorphism between \'etale cohomology and 
the singular  cohomology of complex algebraic varieties}

There is a canonical comparison isomorphism

 H_top( X(C), Q ) \tensor Q_l == H_et ( X, Q_l )

for X a separated scheme over C of finite type (cf. [Katz, SGA4, XVI, 4.1]). 

In other words, a choice of a connected locally connected locally compact topology
on an algebraically closed field K with continuous field operations,
gives rise to a _Q-form_ of \'etale cohomology
 
(*)  H_top( X(K)_top, Q) \tensor Q_l == H_et ( X, Q_l )

(A field with such a topology is necessarily isomorphic to C or R.)
 
Let K be an algebraically closed field of characteristic 0.
Let Var/K denote the category of smooth projective separated 
schemes of finite type over K, 
and let Mot/K be the category of pure motives 
according to one of the definitions in [Kleiman];
there is a canonical functor Var/K --> Mot/K. 
Let Q-Vect and Q_l-Vect denote the category of 
finite dimensional vector spaces over Q, resp. over Q_l.


Let H_et(-,Q_l) denote the functor of \'etale cohomology
which sends a variety X into the Q_l-vector space 
H_et^0 (X,Q_l) \oplus ... \oplus H_et^{2 \dim X}(X,Q_l). 


Conjecture(Categoricity for comparison isomorphisms): 
  Let H: Var/K ---> Q-vect be a functor which admits an isomorphism
    H( X, Q ) \tensor Q_l == H_et ( X, Q_l ).
  
  Assume both H and H_et factor via the category Mot/K 
  of pure motives over K. 
 
  Any isomorphism 
      H( X, Q ) \tensor Q_l == H_et ( X, Q_l )
  comes from topology, 
  i.e. there is a natural transformation 
     H ==> H_top ( X(K)_top, Q)
  which induces identity on H_et ( X, Q_l ) via the isomorphisms
    H( X, Q ) \tensor Q_l == H_et ( X, Q_l ) == H_top( X, Q ) \tensor Q_l .

Remark:
   Standard conjectures [Kleiman,\S1,p.6/4] imply 
   that Mot/K is well-defined, is a semi-simple Abelian category,
   and that that both the singular and \'etale  cohomology functors 
   factor via the category of pure motives.


# Related more explicit conjectures

We may restrict H and H_et to a smaller source subcategory of Var/K
and hope to formulate related conjectures which are more explicit and are 
easier to handle. Below we do so and consider H and H_et restricted
to a subcategory generated by a single Abelian variety.

This is done as follows.

Etale cohomology is a functor from schemes and therefore
are functorial wrt field automorphisms 
sigma : K--->K, eqv. sigma : Spec K ---> Spec K.

This lets to express "comes from topology" in terms
of Aut(K)-action on the \'etale cohomology, i.e. 
isomorphisms
  H_et ( sigma(X), Q_l ) == sigma_* H_et( X, Q_l) . 

Further, it is convenient to think of a comparison isomorphism
as a subfunctor H of H_et such that 
each $\Q$-vector space $H^i(X) \subseteq \Het^i(X)$ is dense and 
is of the same dimension.




Conjecture (Categoricity of comparison isomorphisms):
  A subfunctor H of H_et  comes from  a comparison isomorphism with 
  the singular cohomology, 
   i.e. is of form 
         H_top( X(\sigma(C)), Q ) \subseteq H_et ( X, Q_l ) ,
   iff   
    each Q-vector space $H^i(X) \subseteq \Het^i(X)$ is dense and has the
    same dimension.

 
Conjecture A:
    The Aut(H_et)-orbit of H_top is consists of those subfunctors $H$ of
    H_et which are ``Q-forms of H_et'',
    i.e. each Q-vector space $H^i(X) \subseteq \H_et^i(X)$ is dense and has the
    same dimension.



Conjecture B: 
    Restrict H_et and H_top to the subcategory of varieties defined 
    over a number field k,  or a subcategory consisting of Cartesian powers 
    of a single variety A. 
 
 Then Q-forms of H_et form at most finitely many $Aut(Qbar/k)$-orbits.


## Partial results


Remark/Expectation:
    If H is a Q-form of H_et, then \Aut(H) is a pro-algebraic group
    over Q, and \Aut(H) \tensor_\Q \Q_l ~= \Aut(H_et).

    Perhaps some sort of converse also holds?

Example:
    Conjecture A holds for $A$ an Abelian variety, $\dim A=g$, such that the
    Mumford-Tate group is $\Symp_{2g}$.
Proof (sketch):
    The Weil pairing corresponds to a motive, 
    and by functoriality we have that, 
    for a $\Q$-form $H$ of $\Het |_{<A>}$,
    the non-degenerate Weil pairing
        $\omega : (\Het^1(A))^2 -> \Het^0(A) ~= \Q_l$
    restricts to a pairing
        $\omega : (H^1(A)^2 -> H^0(A) ~= \Q$,
    which is easily seen to be non-degenerate.

    Now let $H_i$ be a $\Q$-form for $i=1,2$.

    Let $(x_i^1,...x_i^g,y_i^1,...,y_i^g)$ be a symplectic basis for $H_i^1(A)$.
    Then each is also a symplectic basis for $\Het^1(A)$,
    and so some $\sigma \in \Symp(\Het^1(A),\omega)$ maps $H_1^1(A)$ to $H_2^1(A)$.
    The assumption on the Mumford-Tate group precisely means that such a
    $\sigma$ extends to $\sigma \in \Aut(\Het |_{<A>})$,
    and it follows from the fact that the cohomology of an Abelian variety is
    generated by $H^1$ that $\sigma(H_1) = H_2$
    (TODO: details).


Conjecture (a generalisation of the example):
    Assume the Mumford-Tate group G=Aut(H_et|<A>)  has the following property:
        if V_1 and V_2 are Q-vector subspaces of H_et(A,Q_l)
        such that GL(V_i) /\ G(Q_l) is dense in G(Q_l) for i=1,2,
        then there is a g in G(Q_l) such that gV_1=V_2 (setwise).

    Then A  holds for H_et|<A>.


# Characterising the orbit of $\Htop$ under Galois automorphisms

Let $X$ be a projective algebraic variety over a number field $k_0$.

It is conjectured that the image of $\Gal(k_0)$ in the motivic 
Galois group MT is open, and it is furthermore conjectured that 
$\Aut(\Het |_{<X>}) = MT(\Q_l)$; see [Serre, 2.1?, 9.1?].

If also Conjecture A holds for $\Het |_{<X>}$, it follows that up to the
action of $\Gal(k_0)$, there are only finitely many subfunctors $H$ of 
$\Het |_{<X>}$ such that each $H^i(Y) <= \Het^i(Y)$ is dense and has the same
dimension. It is important here that the source category is a category of
(pure) motives.

Remark:
 There are similar conjectures for finite adels instead of Q_l, 
 cf. [Serre, 11.4?(ii), 11.5?], cf. also [Serre, 10.2?, 10.6?]







## Categoricity for Betti cohomology of algebraic varieties

# A naive categoricity conjecture for Betti cohomology

A naive conjecture is that the following description is categorical:

Conjecture (a naive vague conjecture on Categoricity of Betti cohomology):
  The following description is categorical: 
  a Weil cohomology theory with Q-coefficients which factors 
  via the category of pure motives.


That is, a Weil cohomology theory with Q-coefficients which factors 
via the category of pure motives, necessarily has form 
H_top ( X_sigma(K), Q ) for some field isomorphism sigma:K-->C.  

Note, however, that the conjecture is vague, as the notion of 
"isomorphism" ("be of form") of cohomology theories is not specified.



## Base Purity Conjectures


Conjecture:
   The field is purely embedded into the structures
   corresponding to functors

   (i)     H:Var/Qbar ---> k-dga
   (ii)    H:Var/C ---> k-dga
   (iii)   H:Var/Q ---> Mot/Q

Moreover, the structure (ii) is an elementary extension of (i) and
the cohomology ring H(V) is definable for every variety over C.


It is tempting to assume that 
Lefschetz weak and strong theorem [Kleiman,\S4,p.11/9],                                          
and the Lefschetz standard conjecture [Kleiman,\S4,p.13/11],    
may be used to prove definability of the cohomology ring
of a hyperplane section. 

Several of the Standard Conjectures [Kleiman, \S4,p.11/9] claim that 
certain cohomological cycles (construction) correspond to algebraic 
cycles. This feels related to many of the conjectures above, 
in particular to the purity conjectures.


TODO:

 1. Define a model-theoretic structure and language corresponding
 to the notion of a Weil cohomology theory, and formulate 
 a categoricity conjecture hopefully related to the Standard Conjectures ([Groth, Kleiman])
 and conjectures on the motivic Galois Group and related Galois representations [Serre].

 2. Do the same in the language of functors, namely:
   2.1. Consider the family of cohomology theories on Var/K coming
   from a choice of isomorphism K=~C.
   2.2. Define a notion of isomorphism of these/such cohomology theories,
    and  what it means to a "purely algebraic" property of such a theory.
   3.3. Find a characterisation of that family up to that notion of isomorphism
   by such properties. Or rather, see it is equivalent to a number of well-known conjectures
   such as the Standard Conjectures etc.





References:
                                                                                                                
 [Andre04] Andre, Yves, Une introduction aux motifs (motifs purs, motifs mixtes, periodes).                     
 Panoramas et Syntheses 17. Societe Mathematique de France, Paris, 2004.                                        
                                                                                                                
 [Andre08] Galois theory, motives and transcendental numbers                                                    
 http://arxiv.org/abs/0805.2569                                                                                 
 
 [Grothendieck] Standard Conjectures on Algebraic Cycles, 
   http://mishap.sdf.org/temp/Grothendieck_Standard_Conjectures_on_Algebraic_Cycles.pdf



 [Serre] Properties conjecturales des groupes de Galois motiviques et des representations l-adiques             
 http://mishap.sdf.org/temp/Serre-Motivic-Galois.pdf                                                           
                                                                                                                
 [Kleiman] Standard Conjectures                                                                                 
 http://mishap.sdf.org/temp/Kleiman_Standard_Conjectures.pdf                                                   
                                                                                                                
 [Katz] Review of l-adic cohomology                                                                             
 http://mishap.sdf.org/temp/Katz_Review_of_l-adic_cohomology.pdf                                               
 
  (in Uwe Jannsen, Steven L. Kleiman. Motives (Proceedings of Symposia in Pure Mathematics) (Part 1). 
  American+Mathematical Society (1994) )                                                                                          


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