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\title[Standard conjectures in model theory]{
standard conjectures in model theory, and categoricity of comparison isomorphisms
%Standard conjectures on categoricity
%of \'etale cohomology in model theory}
%of comparison isomorphisms
%in model theory
}
\author[misha gavrilovich]{notes by 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 Problems of Regional Economics 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}
\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 comparison isomorphism of \'etale cohomology/fundamental groupoid
admits such a purely algebraic definition (characterisation). These conjectures are direct analogues of categoricity
theorems and conjectures in model theory, particularly those on pseudoexponentiaton [Zilber].
We then show that
some special cases of these conjectures seem related to Grothendieck
standard conjectures and conjectures about motivic Galois group, particularly the image of $l$-adic Galois representations.
Note that an algebraic geometer might interpret %is likely to understand %interpret
the question differently and in that interpretation, the answer is well-known to be negative.
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{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 describes $H_{\mathrm{top}}$ (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 $l$-adic \'etale cohomology theory. This is an algebraic description in itself.
However, note that it considers the $l$-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
%%$l$-adic \'\'etale cohomology
%%\item the profinite completion of the topological fundamental groupoid functor is \'\'etale fundamental groupoid
%%\end{itemize}
%%
\subsection{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.
\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 it could happen.
\subsubsection{$l$-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 $l$-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 cohomology classes which 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:
$$G_E=\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)$$
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
$\left$ is the least Tannakian subcategory of $\Mot_k$
containing $E$, and ${\Hsing}_\sigma$ is the fibre functor on $\left$
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 $l$-adic Galois representations in $G_E(\Ql)$.
Both say it is dense or open in the group of $l$-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).
%%
\subsubsection{Pseudoexponentiation: automorphisms groups} It is known that certain automorphisms groups
associated with pseudoexp are largest possible in the following sense.
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 with $a$, 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 $(\C,+,\cdot,\exp)$.
\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 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.
\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 $\delta(X)>0$
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 predimention
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 Corjectures, the conjectural theory of the motivic Galois group,
the conjectures on the image of $l$-adic Galois representaitons, Andr\'e-Grothendieck periods
conjecture, Mumford-Tate conjecture, etc.?
\end{enonce}
\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.
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.
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
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
$$
H\tensor_\Q \Zl \xra{\tau} H_l
$$
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 $l$-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 $l$-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 $l$-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 $l$-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 $l$-adic cohomology theory.
%
%%The comparison isomorphism between singular and $l$-adic cohomology of
%complex smooth projective varieties
%% the separated $\C$-schemes of finite type
%%provides an example of a $\Z$-form
%%of the $l$-adic cohomology theory $\Het(-,\Zl)$.
We conjecture that, up to action of $\Aut(\C)$ defined above,
the comparison isomorphism between singular and $l$-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 $l$-adic cohomology theory $\Het(-,\Zl)$:
\begin{enonce}{Conjecture}[$Z(H_{sing},\Hl)$] Up to $Aut(\C)$ action, there is a unique $\Z$-form of the $l$-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 $l$-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 $l$-adic cohomology.
%Assuming Grothendieck's Standard Conjectures and that, in particular, the $l$-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 $l$-adic Galois representations and motivic Galois group.
Assume Grothendieck Standard Conjectures and that, in particular %(?),
the $l$-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 $l$-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}[$Z(\Het,\genn E k)$] Let $k$ be a number field.
Assume Grothendieck Standard Conjectures and that, in particular,
$l$-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{An example: an Abelian variety.} Let us give an example of a particular case of the conjecture
which is easy to prove.
\begin{example}
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 $l$-adic
cohomology theory $\Het(-\tensor \bar k,\Zl)$ restricted to the category $\genn A k$, up to $\Aut(\kbar/k)$.
\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 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 $\Het |_{}$,
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 $\sigma \in \Aut(\Hetl |_{\left})$,
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).
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|) 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|.
Let $A$ be a motive of a smooth projective variety defined over a number field $k$.
Assume the Mumford-Tate group $G=Aut^\tensor(\Het(-\tensor\bar k, \Zl)_{|\left})$
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
$\Het(-\tensor \bar k,\Zl)_{|\left}$.
%$\Hetl_{|\left\tensor\Qbar}$.
% Then $E_{weak}(\Q_l,\left\tensor\Qbar)$ 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
%$E({\Bbb A}^f, \left\tensor_k\Qbar)$
%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 aapear 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 $\left$ 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 motitivic Galois group of a motive $E$ is
$\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)$.
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
$\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)$ 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 $\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql)$ 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 $l$-adic representation associated to $E$}
$$\rho_{k,l}: Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql).$$
\begin{enonce*}{Conjecture}[3.2?] Let $k$ be a number field.
The image
of the $l$-adic representation associated with $E$
is dense in the group $\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql)$
in the Zariski topology.
\end{enonce*}
\begin{enonce*}{Conjecture}[9.1?] Let $k$ be a number field.
The image $$
Im(\rho_{k,l}:
Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql))$$
is open in $\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql)$.
\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
$\Het(-,\Zl): \left\tensor_k\Qbar\lra \QVect$,
up to Galois action.
%%
%%The following conjecture is probably easy to prove.
%%
%%\begin{enonce*}{Conjecture}[$E(\Ql, \left\tensor_k\Qbar)$]
%%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
%%$\Het(-,\Ql): \left\tensor_k\Qbar\lra \QVect$.
%%\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
%$E({\Bbb A}^f, \left\tensor_k\Qbar)$
%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 [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 existance 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{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}
\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"? }
{\em A $\pi_1$-like functor} is a functor %$\pi:\Var_K\lra\Groupoids$
from a category of varieties to the category of groupoids
satisfying (0-3) above.
Note that by (0) a $\pi_1$-functor comes
equipped with a forgetful natural transformation to the functor of $K$-points.
\begin{enonce}{Conjecture}[$Z(\pi_1^{top})$] 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}[$Z(\pi_1,K)$]
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}
\begin{remark} 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{remark}
\begin{remark} 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{remark}
\def\sepp{\mathrm{sep}}\def\Spec{\mathrm{Spec}\,}
\begin{remark} 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{remark}
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. %(>fixme?: weird wording).
In the next subsection we list several partial positive results.
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 $X^0$ to be 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 $[\mathrm{BH^2K^2 14}]$ probably allow to replace $\C$ by 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},
probably reduce the remaining part of the conjecture
to properties of complex analytic topology
and normalisation of varieties.
\begin{enonce}{Conjecture}[$Z(\pi_1,\Qbar\subset \C)$]
%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 category 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, 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 led to several partial positive results
about 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.
Below, we list several known results, translated from categoricity theorems available in model theory
literature. We list the corresponding category $\VV$ and additional properties requires
of the functors in the family.
\begin{enumerate}
\item \label{ref:Gm} [BaysZilber,Th.2.1] $\char K=0$, $\gen {K^*}$ %$\VV$, $\Ob\VV=\{(K^*)^n\,:\,n\geq 0\}$ is the full subcategory of algebraic tori $(K^*)^n$, $n\geq 0$
\item\ [BaysDPhil, Th.4.4.1; GavrK, Prop.2] $\char K=0$, $\genn E K$ where $E$ is an elliptic curve defined over a number field $k$
with a $k$-rational point $0\in E(k)$;
$\pi(E,0,0)\approx \Z^2$; a finite family
% 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\ [BaysZilber,Th.2.2] $\char K=p>0$, $\gen {K^*}$;
%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
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}
\item\ [BaysDPhil, Th.4.4.1] \label{ab:cat} $\char K=0$, $\genn A K$
%$\VV$ is the full subcategory whose objects are $\Ob\VV=\{A^n\,:\, n\geq 0\}$
where $A$ is an Abelian variety defined over a number field $k$ with a $k$-rational point $0\in A(k)$,
\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}
\item\ [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 $\Ob\VV=\{V^n\,:\, n\geq 0\}$
where $V$ is 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 $n>0$;
recall a group $G$ is subgroup separable iff for each finitely generated subgroup $H0$
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 item (5), 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 $U$.
To prove item (\ref{w-cat}) we use that these two topologies are similar and nicely related. In particular,
\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}
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{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}
This is related to the following geometric fact [GavrDPhil, V.3.3.6, V.3.4.1]:
\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}
In fact we 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.
%%{\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.
\begin{thebibliography}{10}
\bibitem[Andre04]{Andre04}
\newblock Andre, Yves.
\newblock Une introduction aux motifs (motifs purs, motifs mixtes, periodes).
\newblock Panoramas et Syntheses 17.
\newblock Societe Mathematique de France, Paris, 2004.
\bibitem[Andre08]{Andre08}
\newblock Andre, Yves.
\newblock Galois theory, motives and transcendental numbers. \url{ http://arxiv.org/abs/0805.2569 }
\bibitem[BaysDPhil]{BaysDPhil}
\newblock Categoricity results for exponential maps of 1-dimensional algebraic groups \& Schanuel Conjectures for Powers and the CIT.
\newblock Oxford, DPhil thesis.
\newblock \url{https://ivv5hpp.uni-muenster.de/u/baysm/dist/thesis/}
\bibitem[BGH14]{BGH14}
\newblock Martin Bays, Misha Gavrilovich, and Martin Hils.
\newblock Some definability results in abstract Kummer theory.
Int. Math. Res. Not. IMRN, (14):3975–4000, 2014.
arXiv:1112.6012v1 [math.LO].
\bibitem[$\mathrm{BH^2K^2 14}$]{BH2K2.14}
\newblock Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kesaala, and Jonathan Kirby.
\newblock Quasi-minimal structures and excellence.
\newblock Bull. Lond. Math. Soc., 46(1):155–163, 2014.
\bibitem[BHP14]{BHP14}
\newblock Martin Bays, Bradd Hart, and Anand Pillay. Universal covers of commutative finite morley rank groups.
\newblock arXiv:1403.4883 [math.LO], 2014.
\bibitem[Bays-Kirby]{Bays-Kirby}
\newblock Martin Bays and Jonathan Kirby.
\newblock Excellence and uncountable categoricity for Zilber’s exponential fields. arXiv:1305.0493 [math.LO], 2013.
\newblock
%\bibitem[BZ]{BZ}
\bibitem[BaysZilber]{BaysZilber}
\newblock Martin Bays, Boris Zilber.
\newblock Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic.
\newblock Bull London Math Soc (2011) 43 (4): 689-702. DOI: \url{https://doi.org/10.1112/blms/bdq131}
\newblock \url{https://arxiv.org/abs/0704.3561}
%\bibitem[BK]{BK}
%\bibitem{BaysKirby}{BaysKirby}
%Martin Bays, Jonathan Kirby.
%Excellence and uncountable categoricity of Zilber's exponential fields.
%http://arxiv.org/abs/1305.0493
%
\bibitem[DawHarris]{DawHarris}
\newblock Christopher Daw, Adam Harris.
\newblock Categoricity of modular and Shimura curves.
\newblock Journal of the Institute of Mathematics of Jussieu (2015) pp. 1-27
\newblock \url{https://doi.org/10.1017/S1474748015000365}
\url{https://arxiv.org/abs/1304.4797}
\bibitem[GavrDPhil]{GavrDPhil}
\newblock Misha Gavrilovich.
\newblock Model theory of the universal covering spaces of complex algebraic varieties.
\newblock Oxford, DPhil thesis.
\url{http://mishap.sdf.org/misha-thesis.pdf}
\bibitem[GavrK]{GavrK}
\newblock Misha Gavrilovich.
\newblock A remark on transitivity of Galois action on the set of uniquely divisible Abelian extensions in $Ext^ 1 ( E (\mathbb Q), \Lambda )$.
\newblock K-Theory (2008) 38:135–152 \url{https//doi.org/10.1007/s10977-007-9015-0}
\bibitem[Grothendieck]{Grothendieck}
\newblock Alexandre Grothendieck.
\newblock Standard Conjectures on Algebraic Cycles,
%\url{http://mishap.sdf.org/temp/Grothendieck_Standard_Conjectures_on_Algebraic_Cycles.pdf}
\bibitem[HarrisDPhil]{HarrisDPhil}
\newblock Adam Harris.
\newblock Categoricity and covering spaces.
\newblock Oxford, DPhil.
\bibitem[ManinZilber]{ManinZilber}
\newblock Yu. I. Manin. With contribution by Boris Zilber.
\newblock A Course in Mathematical Logic for Mathematicians
\newblock Second Edition
\newblock DOI 10.1007/978-1-4419-0615-1 Springer
\bibitem[Katz]{Katz}
\newblock Katz, Nicolas.
\newblock Review of l-adic cohomology.
% \url{http://mishap.sdf.org/temp/Katz_Review_of_l-adic_cohomology.pdf}
\newblock Motives (Proceedings of Symposia in Pure Mathematics) (Part 1).
\newblock American Mathematical Society (1994)
\bibitem[Kleiman]{Kleiman}
\newblock Kleiman, Steven.
\newblock Standard Conjectures.
% \url{http://mishap.sdf.org/temp/Kleiman_Standard_Conjectures.pdf}
\newblock Motives (Proceedings of Symposia in Pure Mathematics) (Part 1).
\newblock American Mathematical Society (1994)
\bibitem[Serre, Exemple]{Serre, Exemple}
\newblock Serre, J.-P.
\newblock Examples de varietes projectifs conjugees non-homeomorphes.
\newblock C. R. Acad. Sci Paris 58 (1964), 4194-4196
\bibitem[Serre]{Serre}
\newblock Serre, J.-P.
\newblock Properties conjecturales des groupes de Galois motiviques et des representations l-adiques.
% \url{http://mishap.sdf.org/temp/Serre-Motivic-Galois.pdf}
\newblock Motives (Proceedings of Symposia in Pure Mathematics) (Part 1).
\newblock American Mathematical Society (1994)
\bibitem[Tent-Ziegler]{TentZiegler}
\newblock Katrin Tent, Martin Ziegler.
\newblock A Course in Model Theory
\newblock Cambridge University Press, 2012. 260 p.
\newblock ISBN: 052176324X, 9780521763240
\bibitem[Zilber]{Zilber} Boris Zilber.
\newblock Pseudo-exponentiation on algebraically closed fields of characteristic zero.
\newblock Annals of Pure and Applied Logic, Vol 132 (2004) 1, pp 67-95.
\newblock \url{https://people.maths.ox.ac.uk/zilber/expf.html}
% (in Uwe Jannsen, Steven L. Kleiman. Motives (Proceedings of Symposia in Pure Mathematics) (Part 1).
% American Mathematical Society (1994) )
\end{thebibliography}
\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 $k$.
$\left$ denote the least Tannakian subcategory of $\Mot/k$
containing the motive of $A$.
Finally, let $\left\tensor\Qbar$ denote the subcategory of $\Mot/\Qbar$
corresponding to $\left$.
\begin{example} $E_{weak}(\Ql, \left\tensor\Qbar)$ 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 $\Het |_{}$,
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 $\sigma \in \Aut(\Het |_{\left})$,
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|) 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|.
Let $A$ be a motive of a smooth projective variety defined over a number field $k$.
Assume the Mumford-Tate group $G=Aut(\Het_{|\left\tensor\Qbar})$
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 $E_{weak}(\Q_l,\left\tensor\Qbar)$ 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 $\left$ 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
$\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)$.
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 $\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)$ 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 $\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql)$ 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 $l$-adic representation associated to $E$}
$$\rho_{k,l}: Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql).$$
\begin{enonce*}{Conjecture}[3.2?] Let $k$ be a number field.
The group $\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql)$
is dense in the Zariski topology in the image
of the $l$-adic representation associated with $E$.
\end{enonce*}
\begin{enonce*}{Conjecture}[9.1?] Let $k$ be a number field.
The image $$
Im(\rho_{k,l}:
Gal(\Qbar/k)\lra \Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql))
$$
is open in $\Aut^\tensor({\Hsing}_\sigma:\left\lra\QVect)(\Ql)$.
\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
$\Het(-,\Ql): \left\tensor_k\Qbar\lra \QVect$,
up to Galois action.
%%The following conjecture is probably easy to prove.
%%
%%\begin{enonce*}{Conjecture}[$E(\Ql, \left\tensor_k\Qbar)$].
%%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
%%$\Het(-,\Ql): \left\tensor_k\Qbar\lra \QVect$.
%%\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
$E({\Bbb A}^f, \left\tensor_k\Qbar)$
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
$\Aut(\Het |_{}) = MT(\Q_l)$; see [Serre, 2.1?, 9.1?].
These conjectures and $E_{weak}(\Q_l,\left)$ imply $E(\Q_l,\left)$.
In other word,
%Indeed, if also Conjecture $E_{weak}(\Q_l,\left)$ holds for $\Het |_{}$,
%it follows that up to the action of $\Gal(k_0)$, there are only finitely many subfunctors
%$H$ of $\Het |_{}$ 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 $\Het |_{}$,
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 $\sigma \in \Aut(\Het |_{})$,
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|) 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|.
# 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
$\Aut(\Het |_{}) = MT(\Q_l)$; see [Serre, 2.1?, 9.1?].
If also Conjecture A holds for $\Het |_{}$, it follows that up to the
action of $\Gal(k_0)$, there are only finitely many subfunctors $H$ of
$\Het |_{}$ 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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