**25**

votes

**3**answers

2k views

### In which ways can the isogeny theorem fail for local fields?

Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \operatorname{Hom}_G(T_l A, T_l B)$ is ...

**22**

votes

**3**answers

1k views

### Products of primitive roots of the unity

Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...

**22**

votes

**1**answer

3k views

### Fontaine-Mazur for GL_1

For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...

**21**

votes

**5**answers

4k views

### Why no abelian varieties over Z?

Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
the set ...

**18**

votes

**1**answer

1k 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 ...

**17**

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 ...

**15**

votes

**4**answers

2k views

### Why do people think that abelian varieties are the hardest case for the Hodge conjecture?

Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...

**15**

votes

**5**answers

2k views

### Can we count isogeny classes of abelian varieties?

Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...

**15**

votes

**2**answers

799 views

### Are there Néron models over higher dimensional base schemes?

Are there NĂ©ron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field?
If not, under what additional ...

**14**

votes

**4**answers

1k views

### Which curves can be found on Abelian varieties ?

We know tha each genus 2 curve is embedded into its degree 1 Jacobian.
Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian ...

**14**

votes

**1**answer

583 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 ...

**14**

votes

**1**answer

424 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 ...

**14**

votes

**0**answers

381 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 ...

**13**

votes

**4**answers

842 views

### Torsion points in Abelian varieties over number fields

Hello,
Suppose $A$ is an Abelian variety of dimension $g$ over a number field $k$. Then using height functions one can show that there are non-torsion points in $A(\bar k)$. This looks like an ...

**13**

votes

**7**answers

2k views

### Examples of rational families of abelian varieties.

I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...

**12**

votes

**2**answers

843 views

### Are Jacobians principally polarized over non-algebraically closed fields?

How does one define the Torelli map $M_g \to A_g$ of moduli stacks? On geometric points a curve maps to its principally polarized Jacobian.
So what I am asking is: if I have a curve $C$ over a ...

**12**

votes

**1**answer

1k views

### isomorphism of abelian varieties

Let $A, B, C$ and $D$ be abelian varieties (over $\mathbb{C}$) such that $A \times B \cong C \times D$, and $A \cong C$. From the irreducibility of abelian varieties, we can say that $B$ and $D$ are ...

**12**

votes

**2**answers

2k views

### Why were Abelian functions so important in the 19th century?

Felix Klein, when discussing how the popularity of areas in mathematics rises and falls, mentions that in his youth Abelian functions were at the summit of mathematics, and that later on their ...

**12**

votes

**1**answer

701 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 $S$, ...

**12**

votes

**1**answer

389 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 ...

**11**

votes

**3**answers

595 views

### Why study CM abelian varieties?

I know that abelian varieties of CM type have central importance in algebraic geomtry and number theory. There are many conjectures and concepts related to them like Andre-Oort, Coleman conjecture, ...

**11**

votes

**1**answer

393 views

### Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties?

If $X$ is a complex Abelian variety of dimension $g$, then
The canonical sheaf is trivial
$\dim {\rm H}^i(X; \mathcal{O}_X) = \binom{g}{i}$.
When $g =1,2$, then any connected, projective ...

**11**

votes

**1**answer

538 views

### Galois action on one-dimensional quotients of l-adic cohomology

Let $A$ be an abelian variety of dimension $g$ over a number field $K$, and $\ell$ be a rational prime. Suppose that the Galois action on the $\ell$-adic cohomology $H^k(A, \mathbb{Z}_\ell) ...

**11**

votes

**0**answers

288 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?

**10**

votes

**2**answers

2k views

### non principally polarized complex abelian varieties

I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an ...

**10**

votes

**2**answers

1k views

### Failure of Theorem of the Cube?

I am trying to understand the theory of cubical structures and am interested in knowing if a disconnected commutative group variety whose identity component is a semi-abelian variety satisfies the ...

**10**

votes

**1**answer

270 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 ...

**10**

votes

**1**answer

1k views

### Crystalline cohomology of abelian varieties

I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...

**10**

votes

**0**answers

295 views

### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...

**10**

votes

**0**answers

158 views

### Infinitely many curves with isogenous Jacobians

Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?

**10**

votes

**0**answers

282 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 ...

**10**

votes

**0**answers

347 views

### About the Bloch conjecture on entire curves

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

**10**

votes

**0**answers

455 views

### Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...

**9**

votes

**3**answers

864 views

### Why can projective varieties just have abelian group operations?

I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is ...

**9**

votes

**4**answers

1k views

### Torsion of an abelian variety under reduction.

Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}_p$. How do you prove that if $q$ is another prime, then the $q$-torsion of $A$ injects into the torsion of $A_p$, ...

**9**

votes

**1**answer

472 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 ...

**9**

votes

**1**answer

408 views

### Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM?

Let $A$ be an absolutely simple abelian variety over a number field $K$. Assume that, for some prime $p$, the Tate module $T_p A$ has a submodule of rank one, invariant under the absolute Galois group ...

**9**

votes

**1**answer

678 views

### When is an Albanese variety principally polarized?

Let (X,x) be a pointed projective variety. Then there exists an abelian variety V which is universal for maps of pointed varieties $(X,x) \to (A,e_A)$, called the albanese variety. When X is a curve, ...

**9**

votes

**4**answers

504 views

### Algebraic cycles of dimension 2 on the square of a generic abelian surface

I would like to know, what is known on algebraic cycles of dimension 2
modulo algebraic or rational equivalence
on the square of a generic abelian surface.
First, let $A$ be a generic abelian surface ...

**9**

votes

**2**answers

493 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 ...

**9**

votes

**0**answers

170 views

### Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...

**9**

votes

**0**answers

151 views

### Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...

**8**

votes

**3**answers

981 views

### “Albanese” schemes: When does an “initial abelian scheme” exist under a given scheme?

For a variety V, its Albenese variety Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety. Can we say something similar for an arbitrary ...

**8**

votes

**5**answers

636 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 ...

**8**

votes

**2**answers

432 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 ...

**8**

votes

**3**answers

584 views

### possible CM-types of abelian varieties

Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End_{\mathbb{C}}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. ...

**8**

votes

**1**answer

658 views

### Quotient of abelian variety by an abelian subvariety

Let $k$ be a field and $A$ an abelian variety over $k$. Suppose that $B$ is an abelian subvariety of $A$. Consider the following fact:
There exists an abelian variety $C$ over $k$ and a surjective ...

**8**

votes

**1**answer

380 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 $S$
...

**8**

votes

**1**answer

354 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 ...

**8**

votes

**1**answer

263 views

### What is the structure of the group of rational points of an abelian variety over a Laurent series field?

Let $K = \mathbb{F}_q((t))$, and let $A_{/K}$ be a nontrivial abelian variety. Then $A(K)$ is a compact $K$-adic Lie group. What can be said about its structure?
By way of comparison, if ...