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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What is the structure of the group of rational points of an abelian variety over a Laurent series field?

Let $K = \mathbb{F}_q((t))$, and let $A_{/K}$ be a nontrivial abelian variety. Then $A(K)$ is a compact $K$-adic Lie group. What can be said about its structure? By way of comparison, if ...
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143 views

Isogeny of abelian varieties

Suppose we have a curve $X$ (of genus $\geq 3$), and we know that $\{\phi_i : X \to E_i\ \textrm{ for } i = 1, ..., r\}$ are covers of degrees $d_i$ (with the $d_i$'s not necessarily all equal), ...
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0answers
126 views

stably birational abelian varieties are isomorphic

Can anybody help me to prove the following result: Proposition. Let $A$ and $B$ be abelian varieties over a field $k$ of characteristic zero. Assume that $A \times \mathbb{P}_k^n$ and $B \times ...
2
votes
1answer
150 views

Dual of a Complex 2-Torus

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
6
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194 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. ...
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1answer
216 views

Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...
10
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0answers
205 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 ...
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vote
1answer
113 views

Degree of isogenies between (semi-)abelian schemes

Let $S$ be a connected noetherian normal scheme of dimension 0 or 1 (i.e. $S$ is a connected Dedekind scheme). Let $f:G\to G'$ be a morphism of semi-abelian schemes over $S$. In their book on Neron ...
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1answer
226 views

Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...
2
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0answers
157 views

Descent theory of line bundles on abelian varieties under isogenies (in char p>0)

I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic. Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...
3
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2answers
179 views

Quotient of an abelian surface by an antisymplectic involution

What can we say about the quotient of an abelian surface by an antisymplectic involution?
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137 views

$k$-isogenies and $k$-subgroups of abelian varieties

Let $k$ be a field of char0, with algebraic closure $\bar{k}$. Let $A$ be an abelian variety over $k$ of positive dimension and let $d\geq 1$ be an integer. Let $S(A,k,d)$ be the set of abelian ...
1
vote
1answer
200 views

Nef divisors on abelian varieties

The following question stems from a question I already asked on MO: Nakai-Moishezon theorem for abelian varieties I would like to prove that if $L_0$ is an ample line bundle on an abelian variety ...
6
votes
2answers
334 views

Image of abelian varieties

Let $k$ be an arbitrary field, and let $\varphi:A\to B$ be a morphism of abelian varieties over $k$. If $k$ has characteristic zero, then $\varphi(A)$ has the structure of an abelian subvariety of ...
2
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0answers
100 views

Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$

The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked. ...
2
votes
1answer
205 views

Purely additive reduction of Jacobian of Hyperelliptic curve

For general, let X be an abelian variety of dimension g. We say that X has 'purely additive reduction' at prime p if the dimension of the unipotent radical of the special fiber of the Neron Model of ...
3
votes
1answer
201 views

Duality for rank one modules over a number ring

Let $K$ be a number field, and $R$ an order of $K$. Consider the category $\mathcal{M}$ of all finitely generated $R$-submodules of $K$. If $X$ is an object of $\mathcal{M}$ such that ...
3
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0answers
153 views

alternate interpretations of Galois action on Tate module

Let $E$ be an elliptic curve over a field $K$, and let $\ell$ be a prime different from the characteristic of $K$. Consider the well-known short exact sequence of etale fundamental groups (geometric ...
7
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333 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 ...
3
votes
1answer
391 views

Nakai-Moishezon theorem for abelian varieties

In Birkenhake and Lange's book, they prove a version of the Nakai-Moishezon theorem for complex abelian varieties that says that if $L_0$ is an ample line bundle on a complex abelian variety $X$ of ...
0
votes
0answers
85 views

Morphisms of Neron models

Let $S$ be a connected Dedekind scheme with field of fractions $K$. Let $A_K$ and $B_K$ be abelian varieties over $K$, and let $A$ and $B$ be the Neron models over $S$ of $A_K$ and $B_K$ ...
3
votes
3answers
362 views

General curves of genus 3 as plane sections of Kummer surfaces

Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the ...
5
votes
0answers
129 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 ...
3
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0answers
184 views

K-theory of categories of group schemes and abelian varieties

Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group ...
5
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0answers
164 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 ...
3
votes
2answers
332 views

Intersection multiplicity in abelian varieties

Suppose $A$ is an abelian variety, $X, Y$ are subvarieties of $A$ of complementary dimension, Does every component of $X \cap Y$ contribute non-negatively to the intersection number?
4
votes
1answer
326 views

Conductor of abelian varieties

Let $A$ be a non-zero abelian variety defined over a number field $F$. Let $v$ be a finite place of $F$, and let $f_v(A)$ be the usual conductor exponent of $A$ at $v$ (defined e.g. on p.500 of the ...
22
votes
3answers
2k views

Products of primitive roots of the unity

Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...
2
votes
2answers
151 views

Ramification in Division field of Abelian Varieties II

This is a follow-up question after this The set-up is almost the same as before, Let $k$ be a number field, $p$ be a rational prime. Let $A$ be an abelian variety over $k$ which has a good ...
3
votes
2answers
293 views

Intuitive meaning of $k$-polarized Abelian surface?

Are there any good way to understand $k$-polarized Abelian surfaces? I am aware that if $A \cong \mathbb{C}^2/\Gamma$ is $k$-polarized, the lattice $\Gamma$ can be taken of the form $$ ...
1
vote
1answer
227 views

Nef classes on abelian varieties in positive characteristic

Thomas Bauer shows in http://arxiv.org/pdf/alg-geom/9712019v1.pdf that for a complex abelian variety a nef line bundle is numerically equivalent to an effective divisor (this is shown in Lemma 1.1). ...
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votes
0answers
86 views

field of definition of abelian varieties with extra endormorphism

Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$. Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$? This is of course what ...
1
vote
1answer
285 views

Can the Albanese map be anything?

Sorry for the vague title. This question is about the Albanese map from the variety $M$ of canonically polarized varieties to the set of abelian varieties. (The variety $M$ is not of finite type...) ...
2
votes
0answers
230 views

a reference for Kummer theory, with proofs ?

What is a standard reference for Kummer theory of semi-Abelian varieties ? I need a complete exposition with detailed proofs. Also in prime characteristic, although I am not sure what the statement ...
5
votes
2answers
443 views

etale cohomology of an abelian variety and its dual

Let $A$ an abelian variety over a field $k$ and $A^{*}$ the dual abelian variety. How can we relate the étale cohomology of $A$ with etale cohomology of $A^{*}$?
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vote
0answers
74 views

Bound for field of definition (vs field of moduli) of an abelian variety

Let $A$ be a principally polarised abelian variety over $\mathbb{C}$ of dimension $g$. Let $K$ be the field of moduli of $A$. Proposition. $A$ has a model defined over an extension $L$ of $K$ such ...
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0answers
82 views

do commutative groups torsors have a point in an Abelian extension of the base field?

Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...
3
votes
1answer
189 views

Simple abelian varieties over non algebraically closed fields.

I was wondering what people would normally mean by a simple abelian variety $A$ where $A$ is defined over a field $k$ that is not algebraically closed. The definition I found in for example on the ...
0
votes
1answer
536 views

When is an ample line bundle on an abelian variety base point free?

So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that if $L$ has no fixed ...
0
votes
1answer
113 views

Ramification in Division field of Abelian Varieties

This might be a very simple question, and that might be the reason that I could not find any reference on this. My question is Let $A$ be an abelian variety defined over a number field $k$, and ...
6
votes
1answer
377 views

Prym varieties as Jacobian varieties

A generic abelian variety of dimension 2 or 3 is a jacobian of a curve. Is there a canonical way to determine a curve whose jacobian is a prym variety of a unramified double cover of a curve of genus ...
0
votes
1answer
88 views

Is the stabilizer of an irreducible subvariety of an abelian variety irreducible ?

Let $A$ be a (semi-)abelian variety over an algebraically closed field $K$, and $X$ be a closed irreducible subvariety. Can $X$ have a non-trivial finite stabilizer ? By stabilizer, I mean the closed ...
11
votes
0answers
221 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?
2
votes
2answers
296 views

Is the moduli space of ppAVs smooth?

Let $A_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth? Since $A_g$ is the ...
6
votes
1answer
696 views

Automorphisms of Generic Abelian Varieties

Automorphism groups of elliptic curves are very well understood. Of course, every elliptic curve has the automorphism $[-1]$ of order $2$. If we are over a (algebraically closed) field, this is the ...
6
votes
1answer
264 views

Local Norm Mapping for Abelian Varieties

Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarrow{N_{L/K}}A(K)$$ I ...
2
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0answers
171 views

About Alexeev and Nakamura's paper “on Mumford's construction of degenerating abelian varieites”

Is there anyone familiar with this paper? It seems to me it contains "some" typos and even some small elementary mistakes, which makes my reading very slow. Of course the key reason for my slow ...
4
votes
1answer
390 views

Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms

Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ? Details: Let $S$ be a scheme and $f:S'\rightarrow S$ a ...
2
votes
2answers
355 views

Moduli Spaces of Higher Dimensional Complex Tori

I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action. Similarly, I have ...
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165 views

Existence of a point on the Shimura variety of PEL-type correponding to a specific abelian variety

I have been puzzle by the following question for a while. Suppose that we have an a Shimura variety $Sh(G,h_0)$ given by some datatum $(L, V, \psi, h_0)$ such as in Section 4.9 of "Travaux de ...