**2**

votes

**1**answer

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

**1**answer

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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 ...

**0**

votes

**1**answer

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)?

**1**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**1**answer

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

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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 ...

**12**

votes

**0**answers

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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

**5**answers

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

**1**answer

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.
...

**5**

votes

**2**answers

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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?

**6**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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 ...

**7**

votes

**3**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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} ...