Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic ...

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possible mumford-tate groups

Consider an abelian variety $A$ over a number field, and look at the representation of its Mumford-Tate group on $H^1(A)$, restricted to the commutator subgroup. Is it possible that every element of ...
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2answers
430 views

The $Pic^0$ of an abelian variety

Given a variety abelian $ A $ defined over an algebraically closed field of characteristic $ 0 $, Mumford define $ Pic^0(A)$= $L \in Pic (A) | T^*_x{L}L = L \ for \ all \ x \ in A$ , where $T_x$ ...
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1answer
306 views

Maps on the identity components of Neron models

Any map $A \to B$ of abelian varieties of the same dimension over a global field $K$ induces a map $\mathcal{A} \to \mathcal{B}$ on the corresponding Neron models over $X$ (where ...
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1answer
528 views

If an abelian variety has an m-torsion point, is the set of all Galois conjugates of the m-torsion all the m-torsion?

I believe this is the case, but I couldn't come up with a proof off the top of my head, so I want to make sure. If $A$ is an abelian variety over some field $K$ (I'm in fact interested only in ...
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1answer
221 views

The Picard group ao abelian varieties

Dear, my question is: If A and B are abelian varieties over an algebraically closed field, then Pic ^ {0} (A x B) = Pic ^ {0} (A) x Pic ^ {0} (B)? Since already many thanks Flavio.
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3answers
863 views

Why can projective varieties just have abelian group operations?

I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is ...
4
votes
1answer
386 views

Tamagawa numbers of abelian varieties and torsion.

Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there ...
5
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0answers
258 views

Why is Pic^0(C) of a curve C a variety?

Let $C$ be an abstract non-singular curve. I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety. Any pointers towards a reference would be appreciated.
8
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1answer
786 views

About the Serre-Tate theorem

It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure ...
4
votes
1answer
352 views

Selmer of an abelian variety versus that of its dual.

What is the precise relationship between the Selmer group of an abelian variety and that of its dual? For instance, does the vanishing of one not imply the same for the other? To fix ideas, let $A$ ...
4
votes
1answer
193 views

isomorphism of extensions by abelian varieties

Let $A$ and $G$ be abelian varieties over $\mathbb{C}$. An element $P$ of $\text{Ext}(A, G)$ is an exact sequence $0 \to G \to P \to A \to 0$, here one can give $P$ the structure of an abelian ...
13
votes
4answers
841 views

Torsion points in Abelian varieties over number fields

Hello, Suppose $A$ is an Abelian variety of dimension $g$ over a number field $k$. Then using height functions one can show that there are non-torsion points in $A(\bar k)$. This looks like an ...
9
votes
1answer
408 views

Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM?

Let $A$ be an absolutely simple abelian variety over a number field $K$. Assume that, for some prime $p$, the Tate module $T_p A$ has a submodule of rank one, invariant under the absolute Galois group ...
5
votes
1answer
593 views

moduli space of abelian varieties of CM-type

Fix a CM-field $K$ of degree $2g$, and a natural number $n$ which is a multiple of $g$. Write $\tau_1, \tau_2, \ldots, \tau_g, \rho \tau_1, \rho \tau_2, \ldots, \rho \tau_g$ for the different ...
8
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3answers
584 views

possible CM-types of abelian varieties

Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End_{\mathbb{C}}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. ...
11
votes
1answer
537 views

Galois action on one-dimensional quotients of l-adic cohomology

Let $A$ be an abelian variety of dimension $g$ over a number field $K$, and $\ell$ be a rational prime. Suppose that the Galois action on the $\ell$-adic cohomology $H^k(A, \mathbb{Z}_\ell) ...
5
votes
0answers
112 views

global units on moduli spaces of abelian varieties

This is a question from a colleague. Let $A_{g,d,n}$ be the coarse moduli space over $\mathbb Z$ (of the moduli stack, or assume $n\ge3$ if you like) of abelian schemes of relative dimension $g,$ ...
25
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3answers
2k views

In which ways can the isogeny theorem fail for local fields?

Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \operatorname{Hom}_G(T_l A, T_l B)$ is ...
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1answer
420 views

references for abelian schemes

Hi, I have a very basic question. I am looking for references explaining how to construct explicitily a Jacobian starting from a curve or examples of projective equations for an abelian scheme. I ...
6
votes
1answer
203 views

What is the closure of product loci in A_g?

Let $A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. For any partition of $g$, one can consider the corresponding locus inside $A_g$ of products of ...
6
votes
1answer
499 views

Tate models for semistable algebraic varieties with mixed reduction over a local field

It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, ...
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0answers
666 views

deformation of abelian varieties

$k$ is a field of characteristic p, $C_k$ is the category of all artinian local rings with residue field an extension of $k$. $A$ is a dim-$g$ abelian variety over $k$, $L$ is a CM field with ...
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1answer
485 views

$2$-torsion line bundles on abelian varieties

Let $\mathcal{A}_{g,D}$ be the moduli space of abelian varieties of dimension $g$ and polarization $D$ of type $(d_1, \ldots, d_g)$. Let $\mathcal{M}$ be the moduli space parametrizing pairs $(A, ...
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1answer
701 views

Abelian varieties and Selberg class

Hello everyone, I would like to know whether, assuming Selberg's orthonormality conjecture, it would be possible to establish a "natural" correspondence between abelian varieties and functions ...
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2answers
1k views

Failure of Theorem of the Cube?

I am trying to understand the theory of cubical structures and am interested in knowing if a disconnected commutative group variety whose identity component is a semi-abelian variety satisfies the ...
5
votes
1answer
737 views

Abelian subvarieties of abelian varieties — reference request

This question may be too naive, in which case I apologise in advance. Anyway, it is a well-known fact (see e.g. Milne's notes) that any abelian variety A has only finitely many direct factors up to ...
6
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1answer
233 views

Can the simplicity of abelian varieities be implied by the reduction

A is an abelian variety over number field K, with simple good reduction at a finite field $\kappa$, can we deduce that $A$ itself is simple?
7
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5answers
2k views

Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent: 1) $X$ is defined over $\overline{\mathbb{Q}};$ 2) There exists a meromorphic ...
2
votes
0answers
146 views

The automorphism group of a particular ppav

Let $E$ be the elliptic curve $y^2=x^3-x$ defined over $\mathbb F_5.$ It is ordinary with $j=-2$ and $End_{\mathbb F_5}(E)=\mathbb Z[i],$ where $i:(x,y)\mapsto(-x,2y).$ So $Aut_{\mathbb F_5}(E\times ...
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2answers
462 views

Shimura datum of family of fake elliptic curves

Suppose we have a PEL type $(H,\phi ,*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and * is a positive anti involution of H; ...
5
votes
1answer
496 views

Rank 2 vector bundle on a product of elliptic curves

Let $E$, $F$ be two complex elliptic curves, and $A=E \times F$. Let us denote by $\pi_E \colon A \to E, \quad \pi_F \colon A \to F$ the natural projections. For all $p \in F$ let us write $E_p$ ...
6
votes
2answers
680 views

About isogenies of abelian varieties

Why it is true that, over an algebraically closed field, any abelian variety is isogenous to a principally polarized abelian variety?
5
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1answer
579 views

an exercise about elliptic surface in Beauville's book

In Beauville's "Complex Algebraic Surfaces", given an elliptic surface $f : X \to C$ with a generic fiber $E$. Then either $\text{Alb}(X) \cong \text{Jac}(C)$ or there is an exact sequence of abelian ...
3
votes
1answer
412 views

isogenies between abelian varieties that induce isomorphisms?

Let $\varphi : A \to B$ be an isogeny between 2 abelian varieties of dimension $g$. Are there known conditions for the $\ker\varphi$ so that this induces an isomorphism between $A$ and $B$? For ...
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0answers
535 views

étale cohomology with values in the $\ell$-torsion of an Abelian scheme

Let $S/\mathbf{F}_q$ be a $d$-dimensional smooth projective variety and $A/S$ be an Abelian scheme. Is there an easy description of $H^0(S, A(\ell)(d-1))$?` ($A(\ell)$ = union of $A_{\ell^n}$)
8
votes
1answer
656 views

Quotient of abelian variety by an abelian subvariety

Let $k$ be a field and $A$ an abelian variety over $k$. Suppose that $B$ is an abelian subvariety of $A$. Consider the following fact: There exists an abelian variety $C$ over $k$ and a surjective ...
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1answer
516 views

Morphism between polarized abelian varieties

Is it true that if an isogeny between two principally polarized abelian varieties respects the polarization, then it is in fact an isomorphism?
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226 views

quasi-trigonal curves

I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...
11
votes
1answer
393 views

Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties?

If $X$ is a complex Abelian variety of dimension $g$, then The canonical sheaf is trivial $\dim {\rm H}^i(X; \mathcal{O}_X) = \binom{g}{i}$. When $g =1,2$, then any connected, projective ...
22
votes
1answer
3k views

Fontaine-Mazur for GL_1

For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...
6
votes
1answer
649 views

Monodromy groups of families of abelian varieties: a reference request

In Serre's letter to Vigneras of 2 Oct 1986, he summarizes a course he's giving in Paris, explaining how to control the image of the mod-l Galois representations attached to abelian varieties. In ...
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0answers
457 views

Automorphic representations attached to abelian varieties

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...
6
votes
1answer
447 views

$p$-torsion in the Mordell-Weil group of Abelian varieties injecting in reduction

Let $K$ be a number field and $\mathfrak{p}$ be a place of good reduction. It is easy to see that the reduction map on prime-to-$p$ torsion $A(K)[p'] \hookrightarrow ...
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votes
1answer
452 views

Betti Cohomology of singular Kummer Surface

Let $A$ be a complex torus of (complex) dimension 2 and $X$ the associated Kummer variety $A/\sigma$, where $\sigma(x)=-x$. I would like to compute the cohomology of $X$ with $\mathbb{Z}$ ...
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3answers
405 views

projective subvarieties of the moduli space of abelian varieties

I know that the fibre of $A_{g,n}$ over $\mathbf{F}_p$ is quasi-projective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for ...
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votes
2answers
452 views

Polarizations on intermediate Jacobians

Let $X$ be a Kahler variety of dimension $n$. For each odd number $2k-1 \leq n$ one can consider the $k$-th intermediate Jacobian, that is, the complex torus $$J^{k}X := \frac{H^{2k+1}(X, ...
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votes
2answers
1k views

Why were Abelian functions so important in the 19th century?

Felix Klein, when discussing how the popularity of areas in mathematics rises and falls, mentions that in his youth Abelian functions were at the summit of mathematics, and that later on their ...
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0answers
309 views

Ext of Tate-modules of abelian varieties

Let $K$ be a local field (in fact, finite extension of $\mathbb{Q}_p$) and let $A$ and $B$ be abelian varieties over $K$. Associated to $A$ and $B$ are the Tate-modules $T_p(A)$ and $T_p(B)$. Both ...
2
votes
0answers
152 views

Factorization of symplectic isomorphisms of abelian varieties

Background Let $A$ and $B$ be two abelian varieties with dual Abelian varieties $\widehat A$, $\widehat B$. An isomorphism of Abelian varieties $f\colon A\times\widehat A\to B\times\widehat B$ ...
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3answers
562 views

Is there an intrinsic way to define the group law on Abelian varieties?

On an elliptic curve given by a degree three equation y^2 = x(x - 1)(x - λ), we can define the group law in the following way (cf. Hartshorne): We note that the map to its Jacobian given by ...