**3**

votes

**0**answers

206 views

### 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 ...

**1**

vote

**2**answers

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 $T_x$ ...

**3**

votes

**1**answer

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 ...

**-5**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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.

**9**

votes

**3**answers

865 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

**1**answer

388 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**

votes

**0**answers

259 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**

votes

**1**answer

787 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

**1**answer

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

**1**answer

194 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

**4**answers

842 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

**1**answer

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

**1**answer

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**

votes

**3**answers

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

**1**answer

538 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

**0**answers

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**

votes

**3**answers

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 ...

**1**

vote

**1**answer

422 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

**1**answer

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

**1**answer

501 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, ...

**2**

votes

**0**answers

667 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 ...

**5**

votes

**1**answer

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, ...

**7**

votes

**1**answer

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 ...

**10**

votes

**2**answers

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

**1**answer

739 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**

votes

**1**answer

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**

votes

**5**answers

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

**0**answers

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 ...

**1**

vote

**2**answers

463 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

**1**answer

497 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

**2**answers

682 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**

votes

**1**answer

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

**1**answer

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 ...

**0**

votes

**0**answers

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

**1**answer

659 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 ...

**1**

vote

**1**answer

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?

**2**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

**8**

votes

**0**answers

458 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

**1**answer

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 ...

**2**

votes

**1**answer

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 $\mathbb{Z}$ ...

**1**

vote

**3**answers

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 ...

**3**

votes

**2**answers

453 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, ...

**12**

votes

**2**answers

2k 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 ...

**1**

vote

**0**answers

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

**0**answers

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 $f\colon A\times\widehat A\to B\times\widehat B$ ...

**6**

votes

**3**answers

563 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 ...