## Complex isomorphism class of abelian varieties and $L$-functions

In his famous Mordell paper, Faltings proved that two abelian varietes $$A_1, A_2$$ defined over a number field $$K$$ are isogenous if and only if the local $$L$$-factors of $$A_1, A_2$$ are equal at every finite place of $$K$$. Moreover, there is an analogue of the Birch/Swinnerton-Dyer conjecture for abelian varieties over number fields which predicts, among other things, than the $$L$$-funciton of $$A/K$$ “knows” the Mordell-Weil rank of $$A(K)$$, since it is equal to the order of vanishing at $$s = 1$$ of the $$L$$-function. Further, in the same paper Faltings proved that it suffices to show that for all but finitely many places $$v$$ of $$K$$, the local factors agree.

Thus, the $$K$$-isogeny class of $$A/K$$ can be seen to ‘know’ important arithmetic information about $$A$$, and in the case when $$A$$ is the Jacobian of a curve $$C$$, also important information about $$C$$.

However, we know that there exist many examples (elliptic curves, say) of abelian varieties $$A_1, A_2$$ which are isomorphic over $$\overline{\mathbb{Q}}$$ (for example, quadratic twists of curves) but not isomorphic over their field of definition $$K$$, such that $$A_1(K), A_2(K)$$ have different ranks. For example, it is known that for any elliptic curve $$E/\mathbb{Q}$$, $$E$$ has many quadratic twists with rank at least two. Starting from a curve with rank 0 say, this produces many examples of elliptic curves that observe the aforementioned phenomenon. Therefore, the $$\overline{\mathbb{Q}}$$-isomorphism class of an abelian variety $$A$$ need not see much of the arithmetic over $$K$$.

Nevertheless, if $$A_1, A_2$$ are isomorphic over $$\overline{\mathbb{Q}}$$ then they will become isomorphic over some finite extension $$M/K$$. Thus, the $$L$$-series of $$A_1, A_2$$ over $$M$$ ought to become identical, since they will then be isogenous over $$M$$.

It thus seems that the $$L$$-function of an abelian variety $$A/K$$ depends subtly on the field $$K$$ itself. The question is, for a given abelian variety $$A/K$$, how does the $$L$$-function $$L_M(A,s)$$, given by viewing $$A$$ as an abelian variety over $$M/K$$ for a finite extension of $$M$$, behave as a function of $$M$$?

## A criterion for abelian $C^*$ algebra

Let $$A$$ be a unital $$C^*$$ algebra such that for any two positive elements $$x,y$$ in $$A,$$ whenever $$x\leq y$$ we have that $$x^2\leq y^2.$$ Prove that $$A$$ isabelian.

## Poisson Summation Formula for Square-Integrable Functions on Locally Compact Abelian Groups

The Poisson Summation Formula (PSF) is most often stated with the requirement that the functions in question be in $$L^{1}$$. However, after doing some searching, I found that there is a paper by R.P. Boas, Jr. which extends the PSF to square-integrable functions in the case of $$L^{2}\left(\mathbb{R}\right)$$. As such, I was wondering where I might be able to find a generalization of this result to functions defined on $$L^{2}\left(G\right)$$, where $$G$$ is an arbitrary locally compact abelian group.

If it helps, I’m only really interested in the case where $$G$$ is $$\mathbb{Q}$$ or $$\mathbb{Q}/\mathbb{Z}$$—or, equivalently, their duals, $$\mathbb{A}$$ (the $$\mathbb{Q}$$ adeles) and $$\overline{\mathbb{Z}}$$ (the profinite integers).

## Functors on the category of abelian groups which satisfy $F(G\times H) \simeq F(G)\otimes_{\mathbb{Z}} F(H)$

Edit: According to the comment of Todd Trimble, I revise the question.

What are some examples of functors $$F$$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\simeq F(G)\otimes_{\mathbb{Z}} F(H)$$

## Polarization type of the complement abelian subvariety

Assume that $$P$$ is a Prym variety of a ramified double cover (hence not principally polarized). Let $$A,B\subset P$$ a complementary pair. Assume that the type of the polarization of $$A$$ is given by $$\delta$$ and that $$dim A < dim B$$.

Can one deduce the type of the polarization on B?

If $$P$$ were pp,this is well known.

## A finite Abelian group of prime-power order is an internal direct product of cyclic groups.

A finite Abelian group of prime-power order is an internal direct product of cyclic groups.

## Seeking for a counter-example for regularity properties of abelian categories

Can one construct a Grothendieck category which has enough projective objects, which is also locally finitely generated (or maybe even locally noetherian), but which does not have enough finitely generated projective objects?

More generally, a lot of questions on independence of some regularity/finiteness properties of abelian categories are very natural, but hard to answer. If anyone knows a somewhat systematic reference with (counter-)examples and other answers about this (a lot are spread in the literature, but the books or articles that I know on this topic contain only very few things and are focused on other aspects of abelian categories), I will be very interested!

## Abelian groups having no finite subgroup

My question is:

Are there currently any classification (up to isomorphism) of infinite abelian groups having no non-trivial finite subgroup?

It seems many have asked whether an infinite group can have all proper subgroup being finite, but few are interested in the infinite groups without any non-trivial finite subgroup.

I know such group is equivalent to that no non-trivial element is of finite order and the basic examples are $$\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$$ and their direct sums, with addition (+) as the group operation. However, what are the other examples and do we know everything about them?

Thanks.

## Why is the triangulated category of motives easier than the abelian one?

There are several expository articles with the title “You could have invented [insert something mysterious here]” (a notable one being about spectral sequences, possibly it even started this genre). This question is somewhat similar in spirit to them.

Here it is stated that “Deligne first suggested that it might be easier to define the derived category $$DM(S,\mathbb{Q})$$ of the hypothetical abelian category of mixed motives.” First time I heard about this, it seemed a little bit counterintuitive to me. We were doing some abelian stuff, why should passing to the derived category make anything any simpler? Of course, it is easy now to point to the success of Voevodsky and others and say that it was totally obvious.

The question is assuming you never heard about Voevodsky, Morel, etc., you are in the 1960’s, how could you arrive at the idea that the triangulated category is easier to construct than the abelian category of mixed motives?

## dual abelian scheme (relative Picard functor) vs. Ext sheaf

Let $$A$$ be an abelian scheme over some base scheme $$S$$. Let $$A^\vee$$ be the dual abelian scheme, defined as $$\text{Pic}^0_{A/S}$$ where $$\text{Pic}_{A/S}(T)=\text{Pic}(A_T)/\text{Pic}(A)$$. (maybe some assumptions are needed for this actually to be an abelian scheme again.)

My main question:

1. I “know” that $$A^\vee$$ can be described as (or is isomorphic to) $$\underline{\text{Ext}}(A,\mathbb G_m)$$, at least that’s what I’ve read quite a few times. I’ve never seen a proof and I don’t know a reference. I don’t even know how to define $$\underline{\text{Ext}}(A,\mathbb G_m)$$. Is it the right derived functor of $$\underline{\text{Hom}}(A,\mathbb G_m)$$? Could someone explain the mentioned isomorphism or give a reference?

my follow up questions (which might be superfluous depending on the answer to my first question)

1. Is it possible to define $$\underline{\text{Ext}}(A,\mathbb G_m)$$ in a more explicit way? What is $$\underline{\text{Ext}}(A,\mathbb G_m)(T)$$ for an $$S$$-scheme $$T$$? Is there a relation between $$\underline{\text{Ext}}(A,\mathbb G_m)(T)$$ and $$\text{Ext}(A_T,\mathbb G_{m,T})$$ or $$\text{Ext}(A(T),\mathbb G_m(T)$$? (not necessarily equality but maybe a morphism between 2 of them?) (I was hoping to deduce the finiteness of $$\text{Ext}(A_T,\mathbb G_{m,T})$$ if $$A^\vee(T)=\underline{\text{Ext}}(A,\mathbb G_m)(T)$$ is finite)

2. If 2. is too difficult: Is it possible to say something about the global sections? What is $$\underline{\text{Ext}}(A,\mathbb G_m)(S)$$?

3. What about base changes? The relative Picard functor behaves well w.r.t. base changes. For example if $$A_K$$ is the generic fiber of $$A$$ then the generic fiber of $$\text{Pic}^0_{A/S}$$ is $$\text{Pic}^0_{A_K/K}$$. What happens on the corresponding “Ext-side”? The generic fiber of $$\underline{\text{Ext}}(A,\mathbb G_m)$$ should be $$\underline{\text{Ext}}(A_K,\mathbb G_m).$$ Is the latter just the corresponding tensor product $$\underline{\text{Ext}}(A,\mathbb G_m)\otimes K$$ of $$O$$-modules?

Any help is appreciated, I don’t expect anyone to answer all questions, of course.