Questions tagged [abelian-varieties]

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 geometry and number theory.

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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 ...
unknown's user avatar
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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 ...
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Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$? Next Q:...
Pierre's user avatar
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Another question related to the isogeny theorem for elliptic curves

I was reading the following question: About isogeny theorem for elliptic curves and was interested in the following statement at the end of Torsten Ekedahl's answer: "Note also that the situation is ...
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isogeny of elliptic curves

Let $E$ and $F$ be two abelian varieties of dimension 1 over $\mathbb{C}$. Let $f : E \to F$ be a surjective homomorphism of abelian varieties ($f(0) = 0$). If $\ker (f) \cong \mathbb{Z}/2\mathbb{Z} ...
Tuan's user avatar
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Are abelian varieties degree two covers of some projective space

Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$. There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question. Does there exist a finite morphism $A\to \mathbf{P}^...
Harry's user avatar
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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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Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differential forms

Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and $J$ be its jacobian (defined over $k$). Let $\sigma: C \rightarrow C$ be a $k$-automorphsm of $C$. This automorphism $\sigma$ ...
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Abelian varieties corresponding to Hodge substructures

In an exercise of Voisin book, says: Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set $H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$. ...
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What is the involution on the moduli space of genus 3 curves induced by the Torelli map

Let $M_g$ be the moduli space of genus $g$ curves, $A_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli ...
Asvin's user avatar
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Can an abelian variety dominate a variety of general type?

Let $X$ be a projective (not necessarily smooth) normal variety of general type over $\mathbb{C}$. Let $A$ be an abelian variety and let $A\to X$ be a surjective morphism. Is $X$ zero-...
Youks's user avatar
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$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties

Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties. Mumford ...
Manoel's user avatar
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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?
Vivek Shende's user avatar
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CM abelian varieties over the rationals

Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding $\iota:L\...
Damian Rössler's user avatar
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Coherent cohomology of an abelian scheme and base change

Let $f\colon A \rightarrow S$ be an abelian scheme of dimension $d$. I would like a reference or an argument for the fact that $R^1f_* \mathcal{O}_A$ is locally free of dimension $d$ and that its ...
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Why is the norm map dual to restriction under Tate local duality?

Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...
Question Mark's user avatar
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1 answer
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Purely inseparable isogeny

How to prove purely inseparable isogeny between two abelian varieties is radical (universally injective)? Purely inseparable morphism means the extension between the two function fields is purely ...
Rick Sanchez's user avatar
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2 answers
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abelian varieties with the same CM type are isogenous

Does anybody have a reference for the following fact? All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$? Here abelian variety with ...
questioner's user avatar
3 votes
2 answers
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Almost Northcott properties for heights of abelian varieties

Let $h$ be a function on the moduli space of abelian varieties of dimension $g$ over $\overline{\mathbf{Q}}$. Let $K$ be a number field and let $g\geq 2$ be an integer. Fix a real number $C$. Does ...
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Stabilizers in abelian varieties are also abelian? reference request

Let $K$ be a field of characteristic $0$ (number fields is a sufficient generality), $A/K$ an abelian variety, and $X\subseteq A$ a closed reduced subscheme. I am looking for a reference for the ...
Lior Bary-Soroker's user avatar
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1 answer
365 views

subvarieties of abelian varieties over number fields

How "special" are closed subvarieties of abelian varieties over number fields? (Dimension 1 is easy.) For example: Are there interesting families of varieties of general type which are not closed ...
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Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves

There is an isomorphism between (rational) correspondences on a curve $C/\mathbb{F}_p$ orthogonal to the "valence zero" ones (i.e. orthogonal under intersection pairing to $\{*\}\times C$ and $C\times ...
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Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...
user51764's user avatar
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2 answers
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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 $$ \begin{bmatrix}...
user2013's user avatar
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1 answer
308 views

Explicit period lattices for abelian surfaces

Given an explicit description (as an intersection) of an abelian surface $A$ is there an algorithm for computing the period lattice of the surface? For the specific examples that I am interested in, ...
Johnson-Leung's user avatar
3 votes
1 answer
191 views

geometrical reducedness of the identity connected component (reference request)

I think there are references for this question, but I didn't find it. We know that for a simple abelian variety $A/k$, the rign $\mathrm{End}^0 (A)$ is a division algebra. One use the fact that every ...
user565739's user avatar
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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 $X=Spec{\mathcal{O}_K}...
Saikat Biswas's user avatar
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Galois cohomology of abelian varieties

Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action. For the first Galois cohomology of $M$, ...
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Product of Abelian varieties with complex multiplication

We take Abelian varieties $A_1, A_2,\dotsc,A_n$ over a number field. If $A_1, A_2,\dotsc,A_n$ have complex multiplication, then does the product $A_1\times A_2 \times \dotsb \times A_n$ have complex ...
OOOOOO's user avatar
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3 votes
2 answers
388 views

Generic Mumford Tate group and algebraic points

I will stick with a concrete example for this question, but it should probably be cast in a more general framework. Let $Sym_g(\mathbf{C})$ be the space of symmetric matrices of order $g$ with ...
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3 votes
1 answer
581 views

Endomorphisms of abelian varieties with real multiplication

Let us work over $\mathbb{C}$ to make life easier. I've came across to the following definition. Let $F$ be a totally real number field of degree $g$, with ring of integers $\mathcal{O}_F$. An ...
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3 votes
1 answer
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Triviality of torsors after a field extension of bounded degree

Let $G$ be an abelian variety defined over a ring $R$. Is there a natural number $n$ such that, for any field $k$ over $R$ and any $G_k$-torsor $T$, there exists an extension $L/k$ of degree $n$ for ...
Rami's user avatar
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3 votes
1 answer
345 views

$\mathbb Q_p$ étale local sytem in characteristic $p>0$

Let $k$ a field with $char(k)=p>0$, separable closure $k^{sep}$ and $f:X\rightarrow Y$ a smooth projective morphism of smooth variety over $k$. 1)Is it true that there exists a (EDIT) dense open ...
Emiliano Ambrosi's user avatar
3 votes
1 answer
136 views

Are the Prym varieties geometrcally nondegenerate subvarieties of the Jacobians?

A subvariety $V$ of an abelian variety $X$ is geometrically nondegenerate if it meets any subvariety of $X$ of dimension bigger than or equal $codim(V)$. My question is about the Prym varieties as ...
Z.A.Z.Z's user avatar
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3 votes
1 answer
350 views

How do I find a smooth curve in $J(C)$ through the 2-torsion points?

Let $X$ be the Jacobian of a genus 2 curve over $\mathbb{C}$. Let $L=\mathcal{O}(nC)$, where n is an even number. Is it possible to find a smooth curve from $|L|$ which is fixed by the involution $x\...
gradstudent's user avatar
3 votes
1 answer
927 views

Construction of Kummer map for abelian variety

Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...
Harry's user avatar
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3 votes
1 answer
591 views

Kuga-Satake with p-adic methods

Is it possible to construct the Kuga-Satake abelian variety attached to a K3 surfaces (over a local field) only using p-adic methods? If the K3 surface is defined over a local field, the Kuga-...
Rogelio Yoyontzin's user avatar
3 votes
1 answer
186 views

Surfaces of general type with $q=1$

Let $X$ be a smooth projective connected surface of general type over $\mathbb{C}$ with $q(X) = 1$, where $q(X) = \mathrm{h}^1(X,\mathcal{O}_X)$. Let $E$ be the Albanese variety of $X$, and let $X\to ...
Pat's user avatar
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1 answer
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What is the geometric quotient of the abelian threefold?

Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and its element $\zeta \neq 1$, $\zeta^3 = 1$. Also, let $E\!: y^2 = x^3 + b$ be an elliptic curve of $j$-invariant ...
Dimitri Koshelev's user avatar
3 votes
1 answer
527 views

Uniqueness of presentation for semi-abelian varieties

Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence $$ 1 \to T \to G \to A \to 1$$ of algebraic groups, where $T$ is an algebraic ...
57Jimmy's user avatar
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3 votes
1 answer
382 views

How to produce low genus curves on abelian surfaces?

I would like to find "simple" complex algebraic curves (i.e. low genus) on a complex abelian surface $A$ (which are not just abelian subvarieties or translates of them). For example, a genus 2 curve, ...
Kim's user avatar
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3 votes
1 answer
151 views

why the division field of an abelian variety contains a cyclotomic field?

Given an abelian variety $A$ defined over $\mathbb{Q}$, for a positive integer (we can suppose prime) $\ell$, let $A[\ell]$ denote the group of points of $A$ that are annihilated by $\ell$, the ...
A. GM's user avatar
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3 votes
1 answer
491 views

Néron models vs integral models

Let $X$ be a smooth projective $k$-scheme, $k$ being a number field. Let $\mathcal{O}_k$ be the ring of integers of $k$. Fix a large enough category of schemes $\text{Sch}/k$ containing $X$, and ...
user avatar
3 votes
1 answer
363 views

Mumford-Tate groups of abelian surfaces

For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases: 1) $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$ 2) $E$ has complex ...
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3 votes
1 answer
376 views

Reduction of Abelian Varieties with Complex Multiplication have Complex Multiplication

Let $A$ be an abelian variety of dimension $g$ over $C$ with complex multiplication by a CM field $K$ where $[K:Q] =2g$. By this I mean that End($A$) $\cong \mathcal{O}_K$. Then, $A$ has a model over ...
user78330's user avatar
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3 votes
1 answer
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Potential good reduction of abelian varieties

In Corollary 3 on page 498 of the article "Good reduction of abelian varieties" it says that, under some specified conditions, the minimal subextension $L/K$ of $\overline{K}/K$ over which an abelian ...
Adam Battelle's user avatar
3 votes
1 answer
1k views

books (or notes) on complex multiplication

This would be a vague question, but I still want to ask here. Do you have any recommended book on complex multiplicaton. I know only 2 books: Shimura's book ...
user565739's user avatar
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upper bounds for the ranks of the minus parts of modular jacobians

Let $p$ be a prime, and $J^-(p)$ be the maximal quotient of the Jacobian of the modular curve $X_0(p)$ on which the involution acts by $-1$. Is anything known or conjectured about upper bounds for ...
Eric Larson's user avatar
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3 votes
1 answer
519 views

Isomorphism on p-torsion of Neron models

Let $A$, $B$ be abelian varieties over $\mathbb{Q}$, with corresponding Neron models $\mathcal{A}$, $\mathcal{B}$ over $X=Spec{\mathbb{Z}}$. Let $p$ be an odd prime of good reduction for both $A$ and $...
Saikat Biswas's user avatar
3 votes
1 answer
261 views

$p$-power torsion of semiabelian variety

Let $K$ be a finite extension field of $\mathbb{Q}_p$. Let us consider a semiabelian variety $G$ defined over $K$, i.e there exists an extension of an abelian variety $B$ and a torus $T$ defined over $...
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