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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the group of all biholomorphic group automorphisms of complex tori

My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group. Let ...
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155 views

Covering of Abelian variety by product of elliptic curves

Let $A$ be an complex abelian variety of dimension $n$. Is it possible to find $n$ elliptic curves $E_1,\dots,E_n$ such that the product $E_1\times \dots \times E_n$ of the elliptic curves etale ...
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89 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 ...
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130 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 ...
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359 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 ...
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348 views

What is the motivation for defining the conductor of an abelian variety?

Let $K$ be a $p$-adic field, and let $A$ be an abelian variety over $K$. The conductor of the abelian variety is often defined as $2u+t+\delta$, where $u$, $t$ and $\delta$ are invariants related to ...
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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 ...
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191 views

Picard number of principally polarized abelian varieties

Let $A$ be an abelian variety of dimension $n$. Over $\mathbb{C}$, at least, it is known that the Picard number (that is, the rank of the Neron-Severi group of $A$) is less than or equal to $n^2$, ...
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266 views

Fourier-Mukai transform for abelian varieties

Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to ...
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159 views

can all CM types be realized by Jacobians?

The question is kind of self contained, but let me develop a bit further. Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is ...
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346 views

Faltings height of a CM abelian variety

Let A be a CM abelian variety, say simple of dimension g, with $End(A) = O_K$, where $K$ is a CM field of degree $2g$. Is there an upper bound for the Faltings height $h(A)$ in terms of the ...
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338 views

Connected cycles of Shimura curves in $A_{g}$ not contained in larger Shimura subvarieties

Is there always a finite family of Shimura curves $(C_{i})$ in $A_{g}$ the moduli space of principally polarized abelian varieties of dimension $g(\geq 2)$, such that the union $\cup C_{i}$ is ...
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110 views

algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights) Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...
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345 views

Minimal polynomial of symmetric endomorphism on abelian variety

Let $(A,\Theta)$ be a principally polarized abelian variety over an algebraically closed field $k$, and let $f$ be a symmetric endomorphism of $A$ (that is, $f^\dagger=f$ where $\dagger$ denotes the ...
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235 views

Picard number of abelian variety [closed]

I would like references or a result about the computation of the picard number of the jacobian of an algebraic curve. What about the special case when the picard number of the Jacobian is one (is ...
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281 views

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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135 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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110 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 ...
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114 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?
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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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182 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 ...
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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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89 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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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 ...
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137 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 ...
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2answers
167 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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128 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 ...
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1answer
183 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 ...
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2answers
309 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 ...
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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. ...
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1answer
186 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 ...
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1answer
195 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 ...
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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 ...
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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
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1answer
357 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 ...
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82 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 $B_K$ ...
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315 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 ...
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123 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 ...
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165 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 ...
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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 ...
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2answers
317 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?
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1answer
279 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 ...
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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 ...
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2answers
133 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 ...
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2answers
235 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 $$ ...
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1answer
214 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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78 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 ...
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1answer
257 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...) ...
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191 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 ...
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358 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 $A^{*}$?