**25**

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 ...

**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 ...

**24**

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 ...

**24**

votes

**1**answer

4k 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 ...

**22**

votes

**3**answers

2k 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 ...

**20**

votes

**3**answers

2k 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 ...

**19**

votes

**4**answers

3k 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 ...

**19**

votes

**1**answer

908 views

### When is “independence of l” known?

My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) ...

**18**

votes

**4**answers

1k views

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

Are there Néron 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 ...

**18**

votes

**4**answers

2k 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 ...

**16**

votes

**2**answers

1k views

### Why polarization of abelian varieties?

Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian ...

**16**

votes

**1**answer

498 views

### Is hyperelliptic cryptography “practical”?

Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...

**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, ...

**14**

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 ...

**14**

votes

**2**answers

1k 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 ...

**14**

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 ...

**14**

votes

**1**answer

650 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

481 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

442 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

1k 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

**3**answers

698 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, ...

**13**

votes

**1**answer

829 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 $S$, ...

**13**

votes

**0**answers

317 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?

**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

455 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 ...

**12**

votes

**1**answer

2k 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 ...

**12**

votes

**0**answers

355 views

### If $k$ is an algebraically closed field of any characteristic, then the fundamental group of $A$ is abelian

This is a followup to my earlier question, see here. I reproduce it as follows.
Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental ...

**12**

votes

**0**answers

265 views

### Counting abelian varieties over finite fields in a given isogeny class

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...

**12**

votes

**0**answers

242 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?

**11**

votes

**1**answer

487 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

930 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 ...

**11**

votes

**1**answer

778 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, ...

**11**

votes

**1**answer

584 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

195 views

### Are Hecke eigenvalues on the cohomology of the Newton polygon strata automorphic?

Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$.
Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus ...

**11**

votes

**0**answers

185 views

### Do all simple factors of jacobians of curves come from correspondences?

For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).
Let $C$ be a curve over ...

**11**

votes

**0**answers

367 views

### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

**11**

votes

**0**answers

280 views

### Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper ...

**11**

votes

**0**answers

467 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 ...

**11**

votes

**0**answers

309 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 ...

**11**

votes

**0**answers

490 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 ...

**10**

votes

**6**answers

2k views

### Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...

**10**

votes

**3**answers

1k 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 ...

**10**

votes

**2**answers

548 views

### Faltings height in short exact sequences

Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height ...

**10**

votes

**1**answer

456 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 ...

**10**

votes

**2**answers

543 views

### On a proposition in Hartshorne's paper “Ample vector bundles on curves”

In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field ...

**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

344 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

211 views

### What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?

It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus).
Is there any similar statement in the tropical case? Naively, the ...

**10**

votes

**2**answers

624 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 ...