# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 Curves'...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) Weil-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 non-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$. In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
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### Property of bundles with connections on abelian variety doesn't hold for additive or multiplicative group?

This question is a followup to two of my previous questions, see here and here. Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using ...
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### Singular curve on an abelian surface

Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 \$...