Algebraic Theta Functions

[gufang zhao]

Does anyone knows the title of the paper on which Martin Olsson gave a series of talks last summer in Zheda?

Best,
Gufang

2009/10/16 tt <sha...@gmail.com>

Some friends of mine opened a website

mathoverflow.net

and you may want to ask/answer math questions on it. Read FAQ there to
learn more about this site. In particular, things like calculus
questions or homework are discouraged; this is not the right place for
that. Enjoy.

[tt]

you can find the pdf file at the bottom of his homepage.

[Pan Xuanyu]

Could TT explain why Stack in arithmetic geometry??What can it solve??Is it any progress in Motive?? Why crystal cohomology??

Best Regards
Xuanyu

--- 09年10月18日，周日, tt <sha...@gmail.com> 写道：

> 发件人: tt <sha...@gmail.com>
> 主题: Re: Algebraic Theta Functions
> 收件人: "Prof.. Shiu-chun Wong's Cocktail Seminar" <cocktai...@googlegroups.com>
> 日期: 2009年10月18日,周日,上午2:21

>
> you can find the pdf file at the bottom of his homepage.
> >
>

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[tt]

For the reason why we want stacks, I guess de Jong should have
explained in the stack seminar, right? But anyway.

I think stack was introduced when studying moduli spaces. You have a
moduli problem (or a moduli functor) and you want to show it is
representable. For instance, the functor that sends a scheme S to the
set of isomorphism classes of elliptic curves over S; it should be
denoted M_{1,1}. But elliptic curves have automorphisms, like
multiplication by [-1], and automorphisms will prevent the functor
from being representable. Then there are two choices. One is to add
level structure to the moduli problem, like considering the set of
isomorphism classes of E/S together with a subgroup of order 3. Then
if no automorphism preserves this extra structure, this new moduli
problem has no automorphism and therefore might be representable (to
show it is really representable one usually embeds the moduli functor
into the Hilbert scheme as a locally closed subfunctor). This new
functor has a natural transformation to the old one, by forgetting the
level structure, and this natural transformation looks like a covering
space map, where the total space is an actual space but the base space
is just a functor. But one can study "geometry" of that functor using
this covering, like computing its cohomology groups as the invariant
of the cohomology of the total space under the action of deck
transformations. The other choice is to promote the old functor into a
stack. Instead of considering isomorphism classes one considers the
category of such objects with isomorphisms between them (so it is a
groupoid). Regarded as a stack (which should be viewed as a geometric
object, like a space), the "covering map" in the 1st choice is really
a covering map. Usually the moduli problem can be solved not only over
the complex numbers but over the integers (or some variants), I guess
that's why it has to do with arithmetic geometry. For instance one can
count "the number of \mathbb F_p-points" of the moduli stack and
define its zeta function, and ask if they are automorphic L
functions. .......For details of the theory, go to the seminar...

We want to say schemes are (or give) mixed motives. But for that to be
true there are lots of conjectures to be proved. If we admit this,
then stacks are also motives. To be precise, the cohomology of a stack
can be computed using some spectral sequence obtained from a smooth
cover by schemes (or algebraic spaces)....
That's why we also expect their L-functions to be automorphic.

It's hard to answer why crystalline cohomology. It has to do with p-
adic Hodge theory, and it gives another realization of motives. When
studying Galois representations, like continuous representations of
the absolute Galois group of \mathbb Q_{p} over a finite dimensional
\mathbb Q_{\ell} vector space, when \ell is different from p things
are much easier. When \ell=p, things are much more complicated....

bst,
tt

[Yi Zhu]

Hi, Shenghao,

1. For the representability problem you mentioned above, I think we also have 3rd appoach by choosing the coarse moduli space. My question is that if one want to study arithmetic problems on stacks, why not studying arithmetics of the coase moduli first? Are there any arithmetic difference among the coverings, stacks and coarse moduli?

Another irrelevant question: In geometric side, geometric properties of moduli of objects "implies" the geometry of the objects. Is there any relation between arithmetics of moduli and arithmetics of objects?

2. Do you know any p-adic Hodge reference?

Best,

Yi

[gufang zhao]

I've got a concrete question.

In Hartshorne's book, in the proof of the fact that an injective sheaf of module $\mathscr{I}$ is flabby(p.207), he claimed $Hom(\mathscr{O}_U,\mathscr{I})=\mathscr{I}(U)$. This is true when $U$ is an affine schemes. But I can't figure out why it is true for general ringed space.

By the way, there's a proof to that proposition by using a flabby resolution of $\mathscr{I}$. But I just want to know whether the identity above is true.

 

Best,

Gufang

[Yi Zhu]

Hi, Gufang,

This is true. Try to think as Hom(A,M)=M, where A is a ring, M is a A-module, Hom is the set of A-module morphisms.

In your setup, all you need is check on each open set. The statement has nothing to do with the scheme structure.

Best,

Yi

[gufang zhao]

I see! Thanks a lot!

 

Best,

Gufang

2009/10/21 Yi Zhu <math...@gmail.com>