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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14
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455 views

Torsion points of abelian varieties in the perfect closure of a function field

The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer. Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...
13
votes
0answers
253 views

Infinitely many curves with isogenous Jacobians

Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians? Does the situation change in positive characteristic?
13
votes
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320 views

Does every Abelian variety have a finite resolution by Jacobians?

One knows that every Abelian variety is a quotient of a Jacobian. Does every Abelian variety have a finite resolution by Jacobians?
12
votes
0answers
276 views

Counting abelian varieties over finite fields in a given isogeny class

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
11
votes
0answers
199 views

Are Hecke eigenvalues on the cohomology of the Newton polygon strata automorphic?

Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$. Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus ...
11
votes
0answers
200 views

Do all simple factors of jacobians of curves come from correspondences?

For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface). Let $C$ be a curve over ...
11
votes
0answers
368 views

Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial? This follows from the conjecture that the maximal ...
11
votes
0answers
289 views

Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s. In the comments of the question, I was directed to the paper ...
11
votes
0answers
492 views

Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties. Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...
11
votes
0answers
312 views

Average ranks of abelian surfaces

Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has ...
11
votes
0answers
498 views

Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
10
votes
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356 views

Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
10
votes
0answers
225 views

Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...
10
votes
0answers
405 views

About the Bloch conjecture on entire curves

The Bloch conjecture states the following: Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim ...
10
votes
0answers
216 views

Mod m versions of the toric part of Tate modules

Let $A$ be a polarized abelian variety over a local field $K$ with residue characteristic $p$. In the course of proving that a polarized abelian variety $A/K$ has semi-stable reduction iff for all ...
9
votes
0answers
331 views

Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property? For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...
9
votes
0answers
165 views

Are automorphisms of abelian varieties detected by the formal group?

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$. Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...
8
votes
0answers
366 views

A frustrating cohomology class on the moduli of abelian surfaces

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...
8
votes
0answers
196 views

Corresponding notion of unramified for motives (or de Rham cohomology)

The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if ...
8
votes
0answers
340 views

Obstructions to deforming an abelian variety

Are abelian varieties unobstructed? That is, given an abelian scheme $X \to \mathrm{Spec}R $ for $R$ a local artinian ring and $\mathrm{Spec} R \to \mathrm{Spec} S $ a nilpotent thickening, can we ...
8
votes
0answers
499 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 ...
7
votes
0answers
216 views

Quadratic twists of 1-motives

Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global ...
7
votes
0answers
237 views

Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$ Is there a (geometrically irreducible) smooth variety ...
7
votes
0answers
275 views

Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s

It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...
7
votes
0answers
403 views

Points of minimum Arakelov height and harmonic arithmetical varieties

Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical ...
6
votes
0answers
164 views

Simplicity of a rank 2 vector bundle over a principally polarized abelian surface

Let $A := \textrm{Jac}(C)$ be the Jacobian of a genus $2$ curve $C$, with principal polarization $\Theta$. Studying some branched covers of $A$, I was led to consider some rank $2$ holomorphic ...
6
votes
0answers
170 views

where do CM abelian varieties get good reduction?

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and assume that $A$ has complex multiplication by the ring of integers of a CM field $K$. Then $A$ has potentially good reduction, that is: ...
6
votes
0answers
233 views

Kernels and cokernels for morphisms of abelian schemes up to isogenies

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. ...
5
votes
0answers
120 views

$p$-adic uniformisation of abelian varieties

In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement: Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...
5
votes
0answers
190 views

Is a semiabelian algebraic space a scheme?

Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is ...
5
votes
0answers
112 views

Effect of Hecke transform on the Mumford-Tate group

Let $Sh_{K}(G,X)$ be a Shimura variety and $Z\subset Sh_{K}(G,X)$ be a special subvariety. $Z$ is given by a Shimura sub-datum $(H,Y)$ with $H\subset G$ an algebraic subgroup which I call the ...
5
votes
0answers
134 views

Compactifying the space of indecomposable abelian varieties

Let $A_g$ be the moduli space of principally polarized abelian varieties and $A_g^0$ the open substack of indecomposable ones. Abstractly we know $A_g^0$ has a compactification with complement a ...
5
votes
0answers
175 views

Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties

Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and ...
5
votes
0answers
272 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.
5
votes
0answers
117 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 $g,$ ...
4
votes
0answers
126 views

Deformations of the moduli space of ppav's

Consider the complex algebraic moduli space $X:=\mathcal A_g^n$ of ppav's of dimension $g$ with some high enough level $n$ structure (so that it represents the corresponding functor). Can one compute ...
4
votes
0answers
254 views

Moduli of coherent sheaves on abelian varieties

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties. We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where ...
4
votes
0answers
313 views

formal group laws of Abelian varieties in positive characteristic

Let $G$ be an algebraic group defined over an (algebraically closed) field $k$. Then one can obtain a formal group law by completing the multiplication map $m: G \times G \to G$ at the unit of $G$. ...
4
votes
0answers
123 views

Are torsion points in a semi-abelian variety over $\mathbb C_p$ bounded?

Let $A$ be a semi-abelian variety defined over (a subfield of) $\mathbb C_p$. Consider its $p$-adic topology with some (non-canonical) metric. Can we bound the distance of torsion points to $0$ with ...
4
votes
0answers
198 views

Does the Albanese map satisfy Torelli's theorem

Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space ...
4
votes
0answers
274 views

Dieudonné modules over rings of charateristic zero

Dear Colleagues, would appreciate if you could recommend references, if such a theory exits, for the following question. Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...
4
votes
0answers
262 views

false elliptic curves and principal polarizations

Hi, Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$. Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
4
votes
0answers
235 views

Good reduction of isogenous abelian varieties over finitely generated fields

Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $A$ and $B$ be abelian varieties over a field $K$, isogenous over some finite extension $L$ of $K$. I want to ask if they have the same ...
4
votes
0answers
268 views

relationship between pairings on principally polarized abelian varieties

Let $X$ be a $g$-dimensional principally polarized abelian variety over $\mathbb{C}$, for example the jacobian of a curve of genus $g$. Let $X = \mathbb{C}^{g}/\Lambda$ where $\Lambda$ is a full ...
4
votes
0answers
743 views

level structures and moduli of abelian varieties

Hello, In the definition of level structure of level $n$ for an elliptic curve $A$, there are two versions: an isomorphism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$. an isomorhpism of ...
4
votes
0answers
589 views

p-divisible groups of superspecial abelian varieties

Let $p$ be a prime and $F$ be an algebraic closure of the field with p elements. I will consider abelian varieties over F up to prime-to-$p$ isogeny. Principal polarizations will be $Q$-homogeneous ...
3
votes
0answers
75 views

How order of divisor with support at infinity is changed at reduction?

Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$. To ...
3
votes
0answers
131 views

Varieties acted upon faithfully by an abelian variety

Let $X$ be a smooth projective variety over the complex numbers. Suppose that some positive-dimensional abelian variety $A$ acts faithfully on $X$. Examples of such varieties $X$ are provided by ...
3
votes
0answers
190 views

Uniruled degenerations of abelian varieties

Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...
3
votes
0answers
114 views

Ampleness of Hodge bundles over complex curves

Let $C$ be a smooth, proper and connected curve over the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its zero-section. ...