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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828 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 ...
2
votes
1answer
280 views

Weil reciprocity on abelian varieties and biextensions?

I was once told, by someone who would likely be right about such things, that the version of Weil reciprocity for abelian varieties (as in Lang, Abelian Varieties) should come out of consideration of ...
2
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1answer
176 views

kernel of an isogeny and coker of its induced map on the Tate module

In a proof in Milne's note "Abelian Variety" (top on p.52), I saw an equality: $ \mathrm{Ker}(\beta)(l) = \mathrm{Coker}(T_{l}(\beta))$, here $\beta$ is an (separable) isogeny of an abelian variety ...
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0answers
198 views

Where can I find a copy of Serre's Cours au college de France 1985-1986?

Hi, I was wondering: where might I be able to find a copy of this work online? And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...
3
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1answer
192 views

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 ...
3
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2answers
378 views

Quotients of Tate modules

Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let $\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over $K$ and denote by $T_{p}A$ its ...
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1answer
138 views

Terminology about Abelian varieties over finite fields

Is there a standard meaning for ordinary and supersingular Abelian varieties over finite fields? If so, where can I find it (together with basic properties about them)?
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1answer
202 views

etale covers of line bundles on an abelian variety

subj: etale covers of line bundles on an abelian variety Is there an explicit decryption of finite etale covers of a line bundle $L$ on an abelian variety and its associated C*-bundles $L^o = L ...
4
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1answer
367 views

Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields

Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields: Type I: totally real, trivial involution Type II and III: ...
2
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0answers
230 views

deRham cohomoloy of CM liftings of Jacobians

Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
2
votes
0answers
140 views

Notation for a canonical quotient of an abelian variety in positive characteristic

This is a light question about notation, but I received no answer in Stackexchange. Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian ...
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0answers
272 views

Weil pairing as an algebraic cycle?

Is there an algebraic cycle corresponding to the Weil pairing on an abelian variety (of dim>1)? Ideally I'd like to see an example as explicit as possible, e.g. an explicitly given variety of dim>1 ...
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vote
1answer
294 views

complex deformations of abelian varieties

Let $A$ be an abelian variety defined over $\mathbf{C}$ (of dimension $>1$) and let $\Theta_A$ be the holomorphic tangent sheaf of $A$. Q: How does one compute $H^1(A,\Theta_A)$ ? If $A$ has ...
4
votes
0answers
247 views

false elliptic curves and principal polarizations

Hi, Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$. Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
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
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0answers
267 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
363 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
515 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
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0answers
176 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
493 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 ...
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0answers
307 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
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1answer
444 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 ...
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0answers
185 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
706 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
439 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. ...
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votes
2answers
401 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
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1answer
402 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 ...
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0answers
254 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?
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votes
1answer
573 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?
9
votes
1answer
270 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
218 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
473 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
195 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
373 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
704 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
302 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
391 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
513 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
251 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
357 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
528 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
296 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
560 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 ...
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votes
3answers
513 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
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0answers
422 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
628 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
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1answer
462 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
462 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
932 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} ...