**3**

votes

**0**answers

333 views

### Decomposition theorem for polarized abelian varieties in positive characteristic

In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...

**2**

votes

**1**answer

448 views

### Tate conjecture for abelian varieties over a finitely generated extension of an algebraically closed field

Let $K$ be a finitely generated extension of an algebraically closed field of characteristic zero, and $A,B$ abelian varieties over $K$.
Then is $Hom_K(A,B)\otimes \mathbb{Z_l} \cong ...

**0**

votes

**0**answers

149 views

### Nef and effective classes on abelian varieties

Is there any characterization of rational nef classes that don't come from effective $\mathbb{Q}$-divisors on abelian varieties? Is there any result along the lines of "Any nef $\mathbb{Q}$-divisor is ...

**0**

votes

**0**answers

191 views

### The formal Group of the dual Abelian Variety

For an abelian variety $A$ with formal group $F$, how is the formal group $F^\ast$ of the dual abelian variety $A^{\vee}$ related to $F$? In general, for a formal group $F$, is there a concept of dual ...

**13**

votes

**1**answer

791 views

### Status of Grothendieck's conjecture on homomorphisms of abelian schemes

In [1] Grothendieck posits the following:
Conjecture. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian schemes over $S$, ...

**2**

votes

**1**answer

229 views

### Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?

Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ldots$, Bombieri and ...

**3**

votes

**0**answers

144 views

### field of definition of isogenies of abelian varieties

Let $A$ be an abelian variety over a field $k$, and let $N$ be a finite subgroup of $A$. Suppose that $N$ is also defined over $k$, or at least that all Galois automorphisms fixing $k$ leave $N$ ...

**0**

votes

**1**answer

163 views

### Reference for Complex Abelian Varieties

I am looking for a reference which explains how theta functions, algebraically independent meromorphic functions, and line bundles all fit together in the context of complex tori. More explicitly, ...

**7**

votes

**0**answers

391 views

### 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 ...

**12**

votes

**1**answer

445 views

### The torsion point count in higher dimension

It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.
If $E/\bar{\mathbb{Q}}$ is an elliptic curve without ...

**4**

votes

**2**answers

208 views

### Theta group representation

Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t_x^*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ ...

**11**

votes

**0**answers

303 views

### Average ranks of abelian surfaces

Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has ...

**2**

votes

**1**answer

421 views

### On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...

**6**

votes

**1**answer

474 views

### Abelian varieties with given endomorphism algebra

I am confused by a statement in the very classical paper of A. A. Albert "On the construction of Riemman matrices II", Ann. Math. 1935, Thm 16. If I understand what he saying, the theorem says that ...

**2**

votes

**0**answers

350 views

### what are the possible CM-fields of PEL type shimura varieties ?

In the paper "Travaux de Shimura" section 6, Deligne had defined a PEL- type shimura variety, for the following datum $(F,E,D,\psi)$, with $F$ a totally real cubic field, and $E$ a imaginary ...

**7**

votes

**1**answer

218 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 ...

**3**

votes

**3**answers

309 views

### the dual abelian scheme

There are plenty of sources discussing dual abelian varieties, but I'm looking for a reference that discusses the construction and properties of the dual abelian scheme. I'm willing to accept general ...

**8**

votes

**0**answers

190 views

### 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 ...

**2**

votes

**0**answers

219 views

### Which curves cut the Hyperelliptic locus?

Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of ...

**6**

votes

**1**answer

937 views

### Poincare line bundle

I am being stuck by the proof of the existence of Poincare 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 ...

**3**

votes

**0**answers

186 views

### Does the Albanese map satisfy Torelli's theorem

Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space ...

**7**

votes

**1**answer

475 views

### For which fields does the isogeny theorem hold

Let $k$ be a field. We say that the isogeny theorem holds over $k$ if, for any abelian variety $A$ over $k$, there are only finitely many $k$-isomorphism classes of abelian varieties $B$ over $k$ ...

**1**

vote

**0**answers

203 views

### How does the line bundles look like on a proper model (or Néron model) of an abelian variety?

How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
Who knows references about this?
In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...

**0**

votes

**0**answers

107 views

### descent of a complex of sheaves

Let X a projective variety over an algebraically closed field on which an abelian variety $A$ acts freely.
Then $X/A$ is a projective variety. Let $f:X\rightarrow X/A$
Let $K\in D_{c}^{\leq ...

**1**

vote

**1**answer

282 views

### complex multiplication

For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \mathrm{dim} (A)$. ...

**4**

votes

**1**answer

354 views

### Properties of subvarieties of a simple abelian variety

Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose ...

**9**

votes

**2**answers

597 views

### What are some consequences of the Mumford-Tate conjecture?

Let $A/K$ be an abelian variety over a number field. On the one hand we have the singular cohomology group
$$V := H^1(A(\mathbb{C}),\mathbb{Q})$$
with respect to some fixed embedding $K \subset ...

**3**

votes

**0**answers

176 views

### Ordinary vs Non-ordinary for GL(2)-type Abelian Surfaces over Q

Let $A_f$ be an abelian surface over $\mathbf{Q}$ of $\mathbf{GL}_2$-type arising from a weight $2$ cuspidal eigenform $f\in S_2(\Gamma_0(N))$. What is known (or expected to be true) for the size of ...

**4**

votes

**1**answer

338 views

### (3,3) abelian surface and k3 surfaces

SOrry for the very specific question, but curiosity bites....
So here's the story: an idecomposable principally polarized abelian surface is embedded in $P^8=|3\Theta |^* $ as a deg 18 surface A. ...

**2**

votes

**0**answers

143 views

### p-divisible group over an algebraically closed field of characteristic p arises from abelian variety

It may be trivially true or trivially false, just a quick ask, if $k=\overline{k}$ and char k = p>0, X is a p-divisible group over $k$, suppose the Newton polygon of $X$ is symmetric, then there ...

**10**

votes

**0**answers

392 views

### About the Bloch conjecture on entire curves

The Bloch conjecture states the following:
Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim ...

**1**

vote

**0**answers

196 views

### Canonical forms for elliptic fibrations with Mordell-Weil group of rank 1 and zero torsion

Consider an elliptic fibration given by the following Weierstrass model:
$$
E: y^2 + a_1 x y + a_3 y =x^3 + a_2 x^2 + a_4 x + a_6,\quad a_6=a_2 a_4.
$$
( I work with characteristic zero).
With the ...

**4**

votes

**0**answers

263 views

### Dieudonné modules over rings of charateristic zero

Dear Colleagues,
would appreciate if you could recommend references, if such a theory exits, for the following question.
Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...

**2**

votes

**0**answers

409 views

### Grothendieck-Messing theory

Hello,
I would like to work out some examples of deformation of isogenies via Grothendieck-Messing theory. Let's take an easy example: Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$ and ...

**5**

votes

**1**answer

328 views

### Genus 2 curves vs Abelian surfaces

In the Satake compactification of abelian surfaces we have the following degeneration of a family of abelian surfaces in $\mathbf{H}_2$
$lim_{t \to \infty}\begin{pmatrix} it & b \\\ b & ...

**3**

votes

**3**answers

442 views

### 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 ...

**7**

votes

**1**answer

431 views

### On simple factors of modular jacobians: endomorphism ring and simplicity of mod p reduction

Let $A_f$ be the abelian variety over $\mathbf{Q}$ arising as a $\mathbf{Q}$-simple factor of the Jacobian $J_0(N)$ of the modular curve associated to a normalized newform $f$ of weight $2$ on the ...

**4**

votes

**2**answers

286 views

### Defining isogenies over smaller fields

I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts ...

**6**

votes

**2**answers

424 views

### 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 ...

**8**

votes

**1**answer

421 views

### Modularity of higher dimensional abelian varieties

In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, one can give a proof using ...

**20**

votes

**3**answers

1k views

### Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...

**3**

votes

**1**answer

235 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, ...

**10**

votes

**1**answer

326 views

### What is the Brauer group of the moduli space of (p.p.) abelian varieties?

What is the Brauer group of the moduli space of principally polarized abelian varieties of a given dimension? I am primarily interested in the "open" moduli space, i.e. not a compactification. The ...

**22**

votes

**1**answer

2k views

### Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic ...

**2**

votes

**1**answer

210 views

### Can we control the size of the intersection of two abelian subfactors of an abelian variety ?

Let $A$ be an abelian variety over an algebraically closed field $K$, and $B$ an abelian subvariety. We know (for instance, from Milne course on AV) that there is an abelian "cofactor" $B'\subset A$ ...

**1**

vote

**1**answer

338 views

### Serre-Tate canonical lifts for finite fields

A result of Serre-Tate states that we can canonically lift an ordinary abelian variety over a perfect field $k$ of positive characteristic to an abelian scheme over the ring of Witt vectors of $k$ and ...

**3**

votes

**0**answers

219 views

### Does semi-stable reduction behave well with Weil restriction of scalars

Let $A$ be an abelian variety over a number field $K$ with semi-stable reduction over $O_K$.
Does the Weil restriction $\textrm{Res}_{K/\mathbf{Q}}A$ of $A$ to $\mathbf{Q}$ have semi-stable reduction ...

**8**

votes

**1**answer

434 views

### Mordell-Weil group of the universal abelian scheme

Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
...

**1**

vote

**0**answers

139 views

### Construction of RM abelian variety from eigenform

Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...

**0**

votes

**2**answers

260 views

### Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dimension smaller than $g$ at some point

Let me be more precise than the title. (This will be my last attempt to do something with abelian varieties. Sorry for all the basic questions. The answers have been great!)
Let $A$ be a simple ...