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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2
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0answers
91 views

Dual Honda systems

Hello, There is an equivalence of categories between p-divisible groups over the ring of Witt vectors $W(k)$ and the category of "Honda systems", that is couples $(M,L)$ formed by a Dieudonné module ...
3
votes
0answers
250 views

lifting abelian varieties

Hello, Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$. This abelian variety is endowed with a principal polarization and an action by $\mathcal{O}_E$, the ring of integers of an ...
3
votes
1answer
343 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 $\mathbf{Q}$. ...
3
votes
1answer
508 views

Algebraic relationships between elliptic functions

Let $f$ and $g$ be tow elliptic function with the same periods, then there exists an algebraic relationship of the form $P(f,g)=0$, where $P$ a polynomial of tow variables and constants coefficients. ...
2
votes
0answers
170 views

CM abelian variety from an algebraic Hecke character?

Hi, Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a "rank 1 CM-motive" $M$ with $\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...
4
votes
1answer
479 views

Deformation space of non-ordinary abelian varieties

It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...
11
votes
0answers
294 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?
2
votes
1answer
441 views

Generalization of singular moduli

$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...
2
votes
0answers
184 views

Question on division field of abelian variety

I am wondering if the following holds or not. Let A be an abelian variety of dimension $d\geq 1$ over $\mathbb{Q}$. Then there is a positive number c depending on d and A such that ...
8
votes
5answers
685 views

Schottky locus in genus 2

Let $\phi_g : \mathcal{M}_g \rightarrow \mathcal{A}_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized ...
2
votes
1answer
432 views

finite non-commutative local group schemes

Can I have some examples of finite non-commutative connected group schemes over a field $k$? I would like also to see some non-trivial torsors over a $k$-scheme $X$ under such group schemes. Thanks. ...
5
votes
2answers
377 views

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 ...
5
votes
1answer
388 views

Mumford-Tate groups of abelian varieties with potentially good reduction everywhere

Let $A$ be an abelian variety defined over a number field, and let $MT(A)$ be its Mumford-Tate group. It is a conjecture of Morita that if $MT(A)$ is anisotropic-mod-center (that is, it has no ...
2
votes
0answers
238 views

modern reference for Néron's “Quasi-fonctions et Hauteurs sur les Varietes Abeliennes”

Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and in English?
6
votes
1answer
541 views

Mumford-Tate group and Galois representations

Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?
8
votes
1answer
258 views

finiteness of torsion points of an abelian variety over a totally real field?

Ribet has shown the following result: if $A$ is an abelian variety over a number field $E$, then the torsion subgroup of $A(E^{cyc})$ is finite. Here $E^{cyc}$ is the union $\bigcup_nE(\mu_n)$, ...
4
votes
0answers
209 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 ...
7
votes
1answer
460 views

To what extent does Poincare duality hold on moduli stacks?

Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the ...
3
votes
0answers
190 views

The Schrodinger representation on the space of sections of a general $(1,3)$-polarized abelian surface

This question arose while I was studying some finite covers of abelian surfaces. Let $(A, \mathscr{L})$ be a $(1,3)$-polarized abelian surface over the complex numbers and consider the vector space ...
1
vote
1answer
364 views

CM liftings of abelian varieties and liftings of Frobenius

It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $ k$ of characteristic $ p>0$ and $ W=W(k)$ is the ring of Witt vectors over $ k$, then the canonical ...
3
votes
3answers
675 views

Elliptic curves on abelian surface

Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?
3
votes
2answers
294 views

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 ...
3
votes
2answers
373 views

CM abelian varieties and potential good reduction

Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite ...
9
votes
1answer
500 views

Can we always find a curve which doesn't have semi-stable reduction

Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...
4
votes
0answers
244 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 ...
2
votes
1answer
339 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 ...
2
votes
1answer
510 views

Serre's open image theorem for products of elliptic curves over function fields via specialization

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325): Let $K$ be a number field and let ...
2
votes
0answers
281 views

canonical bundle of Abelian surface fibrations

For minimal surfaces admitting an elliptic fibration over a smooth curve, there is a famous analysis of possible singular fibers and a canonical bundle formula due to Kodaira. There are two papers of ...
5
votes
2answers
548 views

Is there for every variety X an abelian variety A such that their 1st l-adic cohomologies are isomorphic?

This question is somewhat inspired by Kevin Buzzard's answer to What is the interpretation of complex multiplication in terms of Langlands? and somewhat from my own curiosity about such topics. Let ...
6
votes
3answers
492 views

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 ...
14
votes
0answers
400 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 ...
14
votes
1answer
608 views

Etale endomorphisms of abelian varieties in positive characteristic

Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ${\bf F}_p$ (where $p>0$ is a prime number). My question is : does there exist an abelian ...
3
votes
1answer
181 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 ...
14
votes
1answer
456 views

Characterizations of Abelian varieties (3-folds) in positive characteristic

From this question Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties? I learned that if $X$ is a smooth complex projective variety of dimension ...
3
votes
1answer
452 views

Pulling back a line bundle on the Jacobian to a spin bundle on the curve

I'd like to have an expression for the (or some) line bundle on the Jacobian $J$ of a smooth complex projective curve $C$ with genus $g >1$ which pulls back to a chosen spin bundle (theta ...
1
vote
1answer
878 views

dual isogeny for abelian varieties over a general field

Let $ \phi: A \rightarrow B$ be a separable isogeny between two abelian varieties over a field $k$. One knows that there is a dual isogeny $ \hat {\phi} : B \rightarrow A$ such that $ \hat{\phi} ...
8
votes
2answers
457 views

Has anyone studied the Prym map for double covers with two ramification points?

If $f \colon C \to C'$ is a dominant morphism of smooth projective curves, there is a norm map $f_\ast = \mathrm{Nm} \colon JC \to JC'$ between their Jacobians, and we can consider the abelian ...
3
votes
0answers
559 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 ...
3
votes
1answer
425 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 ...
5
votes
1answer
290 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 ...
5
votes
3answers
500 views

Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields

Let $k$ be a finite field, and $A$, $B$ abelian varieties over $k$. Let $T_p(A)$, $T_p(B)$ be the (contravariant) Dieudonn\'e modules associated to the p-divisible groups attached to $A$ and $B$, ...
2
votes
3answers
701 views

The $ Pic ^ 0 $ of an abelian variety

dear, I need a precise reference to the fact that the connected component of $ Pic (A) $, where $ A $ is an abelian variety over a field $ k $ consists of the following set $\{L \in Pic (A) : T ^*_x L ...
10
votes
0answers
203 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 ...
2
votes
0answers
192 views

structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety

Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?
2
votes
0answers
213 views

Projectivized Normal Cone to Satake Compactification

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$. There exists a compactification, the Satake compactification, which is minimal and has the ...
6
votes
2answers
428 views

Global Sections of the Identity Component of Neron model

Let $A$ be an abelian variety over a number field $K$ and consider the Neron model $\mathcal{A}$ of $A$ over $X=Spec{\mathcal{O}_K}$. If $\mathcal{A}^0$ is the identity component of $\mathcal{A}$, ...
3
votes
0answers
218 views

possible mumford-tate groups

Consider an abelian variety $A$ over a number field, and look at the representation of its Mumford-Tate group on $H^1(A)$, restricted to the commutator subgroup. Is it possible that every element of ...
1
vote
2answers
439 views

The $Pic^0$ of an abelian variety

Given a variety abelian $ A $ defined over an algebraically closed field of characteristic $ 0 $, Mumford define $ Pic^0(A)$= $L \in Pic (A) | T^*_x{L}L = L \ for \ all \ x \ in A$ , where $T_x$ ...
3
votes
1answer
313 views

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 ...
-5
votes
1answer
540 views

If an abelian variety has an m-torsion point, is the set of all Galois conjugates of the m-torsion all the m-torsion?

I believe this is the case, but I couldn't come up with a proof off the top of my head, so I want to make sure. If $A$ is an abelian variety over some field $K$ (I'm in fact interested only in ...