# Tag Info

Accepted

### Embedding abelian varieties into projective spaces of small dimension

Recall that any smooth projective variety of dimension $g$ embeds into $\mathbf{P}^{2g+1}$. Consider now an abelian variety $A$ of dimension $g$ which embeds into $\mathbf{P}^{2g}$. Van de Ven proves (...
• 2,729
Accepted

### When $k = \mathbb{F}_q$ finite field, $X$ always has $k$-rational point, and so $A \simeq X$?

This is a theorem of Lang's from 1956. Here's an online document giving a proof (in the form $H^1(A,k)=0$): Lecture 14: Galois Cohomology of Abelian Varieties over Finite Fields, William Stein. http:...
• 45.1k
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### On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result?

I don't have any contribution for the intuition beyond the fact that, I can't construct something outside the image of (1) so I hope it's surjective. Here is a sketch of the central idea of Tate's ...
• 30.1k
Accepted

### Which hypersurfaces in $\mathbb{P}^n$ are abelian varieties?

Really this is mostly just consolidating what has been (implicitly) said in the comments and cleaning it up a bit (e.g. using the Chow ring instead of singular cohomology), but might as well make it ...
• 5,806
Accepted

### Is hyperelliptic cryptography "practical"?

I am not aware of anybody seriously considering hyperelliptic curves for actual real-world usage, beyond toys, and I would be rather surprised to hear differently from anyone. As you say, ...
• 4,937
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### Is the complement of an affine open in an abelian variety ample?

Welcome new contributor. Yes, that is true. Let $k$ be any field, let $A$ be an Abelian variety over $k$, and let $U\subset A$ be a dense open affine. Denote by $D\subset A$ the complementary ...

### Is every abelian variety a subvariety of a Jacobian?

Embed the dual abelian variety into projective space. Take a smooth hyperplane section and interate until it's one-dimensional, obtaining a smooth curve $C$. By Lefschetz $C$ is irreducible, and the ...
• 129k
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### Tate-Shafarevich group over number fields

No. It is always difficult to "prove" that something is "not known", but this may do: I claim it is not even known when $K=\mathbb Q$, $A$ is an elliptic curve $E$. In fact in this case, the result ...
• 25.5k
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### Points of abelian varieties over purely transcendental extensions

This follows from the following well-known lemma. Lemma. Let $A$ be an abelian variety over $k$. Then any map $f \colon \mathbb P^1 \to A$ is constant. Proof 1. The map $f$ induces a map on the ...
Accepted

### Torsion points of abelian variety as zeros of a section of a vector bundle?

The crucial case is $m=1$: if you have a vector bundle $E$ on $A$ of rank $\dim(A)$ and a section $s$ of $E$ whose zero locus is $\{0\}$, pulling back $(E,s)$ by multiplication by $m$ gives the ...
• 36.5k

• 9,915
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### Which schemes are divisors of an abelian variety?

Any curve of genus greater than two, whose Jacobian $J$ is simple, will do. If it were a divisor on an abelian surface $S$, then there would be a surjection $J\to S$ with positive dimensional kernel, ...
• 2,471

### A geometric definition of the addition law on abelian surfaces

This must be standard, I don't have a reference but the construction is easy: let $y^2=f(x)$ be a genus 2 hyperelliptic curve with $f$ squarefree of degree $5$ or $6$. As a set the Jacobian is the ...
• 10.7k
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### Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?

Here is an example. Let $K = \mathbf{Q}(\sqrt{-21})$. Then the class group of $K$ is $C_2 \times C_2$ and its Hilbert class field is $H_K = \mathbf{Q}(\sqrt{-1}, \sqrt{3}, \sqrt{7})$. In particular, \$...
• 35.1k