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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Non-degenerate points on a Jacobian surface

Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$. Question: Is there a quadratic ...
dfn's user avatar
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Index of the endomorphism ring of an abelian surface

For an abelian surface $A/\mathbb{Q}$ such that $R:=\mathrm{End}_{\mathbb{Q}}(A)$ is an order in a real quadratic field $K$ (so a $\mathrm{GL}_2$-type surface), is there a bound on the index $[O_K : R]...
Sun Ra's user avatar
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flat/crystalline cohomology of abelian variety

Let $A/k$ be an abelian variety over an algebraically closed field and $\ell \neq \mathrm{char}\,k$. In http://jmilne.org/math/articles/1986b.pdf, Theorem 15.1(b) it is proved that $$H^r_{et}(A, R) = \...
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Poincaré line bundle

I am being stuck by the proof of the existence of Poincaré line bundle of complex torus in Griffiths-Harris. Here is the question: Let $M$ be a complex torus and $M'$ be the complex torus dual to $M$...
vu viet's user avatar
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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 ...
JSE's user avatar
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Simultaneous rank jumping of elliptic curves over number fields

Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...
Ariyan Javanpeykar's user avatar
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Monodromy groups that are profinitely dense in Sp(2g,Z)

$\DeclareMathOperator\Sp{Sp}$Assume $g\geq 2$. It is known that there exist finitely generated subgroups of $\Sp(2g,\mathbb{Z})$ of infinite index that surject onto all finite quotients of $\Sp(2g,\...
Gabriele Mondello's user avatar
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Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension

$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...
Maarten Derickx's user avatar
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605 views

Weil pairing and Tate module for $p$-torsion in characteristic $p$

Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$. If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{...
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$p$-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 ...
cannonball's user avatar
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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 ...
Simon Rose's user avatar
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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 ...
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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 $X$...
Masse's user avatar
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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 ...
David Hansen's user avatar
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Canonical liftings of endomorphisms of ordinary abelian varieties

I am looking for a reference to the following ``well known" fact. Let $k$ be a perfect field of prime characteristic $p$ and $W(k)$ its ring of Witt vectors. Let $A_0$ be an ordinary abelian variety ...
Yuri Zarhin's user avatar
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Is the symmetric product of an abelian variety a CY variety?

Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero ...
Harry's user avatar
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2 answers
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Explicit way to construct simple complex tori/abelian varieties of dimension at least 2

The following question was motivated by one of the earliest exercises of Complex Abelian Variaties by Birkenhake and Lange during my presentation last year. It can be shown that any complex torus $X$...
Ehsan M. Kermani's user avatar
7 votes
3 answers
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Families of abelian varieties on the line (or more generally simply connected varieties)

I'm curious whether the following is true: Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...
Alex Youcis's user avatar
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2 answers
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Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?

Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding $\mathcal{M}...
Atsushi Kanazawa's user avatar
7 votes
1 answer
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Galois action on $p$-adic Tate module of Abelian variety over finite field semisimple?

Let $A,B$ be positive dimensional Abelian varieties over a finite field and $p$ be an arbritrary prime. By Zarhin, Homomorphisms of abelian varieties over finite fields http://www.math.nyu.edu/~...
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A "comprehensive" family of abelian varieties

I'm looking for a family of abelian varieties $A\rightarrow S$ over a base that is finite type over $\mathbb{Q}$ (or $\mathbb{Z}$) that is "comprehensive" in the following sense: for every ...
Nathan Lowry's user avatar
7 votes
2 answers
757 views

$p$-torsion of an abelian variety of $p$-rank $0$

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
Lisa S.'s user avatar
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Jacobians defined over smaller fields

Let $L/K$ be an extension of number fields. Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$. In general, the Jacobian $J(X)$ probably doesn'...
Harry's user avatar
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The Poincare Bundle(s) on C \times J

The set up is $C$ is a curve and $J$ is its Jacobian. On the $C \times J$ there is the Poincare bundle $P$ which is the universal family of degree zero line bundles on $C$. For every integer $d$ ...
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Reference request. Finiteness of the Selmer group

Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
Damian Rössler's user avatar
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1 answer
525 views

A constructive proof of the theorem of the cube

Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
Dimitri Koshelev's user avatar
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1 answer
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$2$-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, \...
Francesco Polizzi's user avatar
7 votes
1 answer
180 views

Explicit equations for the universal vector extension of an elliptic curve

The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
Vik78's user avatar
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1 answer
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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 $n^2$, ...
rfauffar's user avatar
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7 votes
2 answers
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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 ...
user42721's user avatar
  • 547
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1 answer
509 views

O-linear Weil-pairing on abelian varieties with real multiplication

Let $A/k$ be an abelian variety with real multiplication by some ring of integers $\mathcal O \subset F$. Let $n$ be an integer prime to the characteristic of $k$. We have the standart $e_n$ pairing $...
Holger Partsch's user avatar
7 votes
1 answer
687 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 A_{\mathfrak{p}}(\kappa(\mathfrak{...
user6960's user avatar
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1 answer
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Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?

Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$. Is it ...
sawdada's user avatar
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7 votes
1 answer
400 views

Weil height of an Abelian Variety with everywhere (potentially) good reduction

Background: Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with everywhere (potentially) good reduction. there are many ways to define the height of $E$, and I will be concerned with the ...
jacob's user avatar
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7 votes
1 answer
326 views

Do complex tori contain quasi-projective open subsets?

Complex tori are not associated to projective varieties in general. But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...
Steven's user avatar
  • 73
7 votes
1 answer
236 views

Automorphic classification of different types of abelian surfaces

For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}_2$ or $\mathrm{Res}_\mathbb{Q}^F (\mathbb{G}_m)$ if it has CM with the imaginary quadratic field $F$. In this case ...
Alireza Shavali's user avatar
7 votes
1 answer
455 views

analogue of Theorem of Mattuck for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$

By a theorem of Mattuck [Abelian Varieties over $p$-Adic Ground Fields, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $A$ of dimension $g$ over ...
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7 votes
1 answer
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Ordinary abelian varieties over a finite field

Let $q$ be a power of a prime $p$. Deligne's paper "Variétés abéliennes ordinaires sur un corps fini" seems to describe an equivalence of categories between ordinary abelian varieties over a finite ...
Calodeon's user avatar
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7 votes
1 answer
228 views

Commutation of endomorphisms of abelian varieties

Let $A$ be an abelian variety over an algebraically closed field $k$. Let $\phi:A\to A$ be an étale isogeny (over $k$). Suppose that the set $\cup_{r\geq 0}({\rm ker}\,\phi^{\circ r})(k)$ is ...
Damian Rössler's user avatar
7 votes
1 answer
464 views

Does the compactified Torelli map extend to a proper map of stacks?

Let $M_g^{ct}$ denote the moduli stack of compact type genus $g$ stable curves and $A_g$ the moduli stack of principally polarized $g$-dimensional abelian varieties. Can someone provide a reference ...
Aaron Landesman's user avatar
7 votes
1 answer
751 views

Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?

I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory. As I understand it, there is a very special class of ...
dorebell's user avatar
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7 votes
1 answer
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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 ...
Sylvain JULIEN's user avatar
7 votes
1 answer
417 views

showing that abelian varieties are de Rham *without* showing that they are crystalline

If $X$ is a smooth projective variety over a $p$-adic field $K$, then Faltings' Theorem says that the etale cohomology of $X_{\overline{K}}$ is crystalline. There have been various steps towards this ...
Tony's user avatar
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0 answers
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Is the universal object over a Hilbert scheme connected?

Hartshorne proved in his thesis that if $S$ is connected, then the Hilbert scheme $\operatorname{Hilb}^p=\operatorname{Hilb}^p(\mathbb{P}^n_S/S)$ is too (where $p\in \mathbb{Q}[z]$). Can the same be ...
Nathan Lowry's user avatar
7 votes
0 answers
217 views

Albanese morphism induces an isomorphism on global $1$-forms

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero equipped with a point $e\in X(k)$. There is Albanese morphism $a:X\to \mathrm{Alb}\,X$ which is initial among pointed ...
SashaP's user avatar
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7 votes
0 answers
149 views

Conditions for $p$-divisible group to come from an abelian variety

Over $\mathbb{Z}_p$ or $\mathbb{Q}_p$, are there any known sufficient conditions on a $p$-divisible group for it to come from an abelian variety?
rj7k8's user avatar
  • 716
7 votes
0 answers
261 views

Factors of the Jacobian of modular curves

Let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$ for p an odd prime. We know that $J_1(p)$ is isogenous to a direct sum of abelian varieties $\oplus_{f}A_f$ where the sum runs over Hecke/...
shehryar sikander's user avatar
7 votes
0 answers
274 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 ...
Cristian D. Gonzalez-Aviles's user avatar
7 votes
0 answers
431 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: ...
conduc's user avatar
  • 71
6 votes
3 answers
787 views

Torsion group of the following elliptic curve

Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$. Can one compute the torsion group $...
user1234's user avatar

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