30 votes
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 (...
  • 2,719
23 votes
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:...
22 votes
Accepted

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 ...
21 votes
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 ...
  • 5,566
18 votes
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, ...
  • 4,303
18 votes
Accepted

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 ...
17 votes

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 ...
  • 122k
16 votes
Accepted

n-th root of unity in n-th division field of abelian variety?

Actually, this is an exercise in Serre's Lectures on Mordell--Weil Theorem: $K(A[n])$ always contains $\mu_n$ if $char(K)$ does not divide $n$ and $A$ is an abelian variety of positive dimension over ...
  • 4,875
16 votes
Accepted

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 ...
  • 25.1k
16 votes
Accepted

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 ...
14 votes

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

A bit of overkill, but it follows from the Weil conjectures. The structure of cohomology ($H^i = \wedge^i H^1$) is computed over the algebraic closure and it follows that the number of points is $\...
14 votes
Accepted

On the moduli stack of abelian varieties without polarization

First, when defining the stack you will have the issue that there are formal deformations of abelian varieties which do not extend to families of abelian varieties over any reduced scheme. These are ...
  • 122k
14 votes
Accepted

Canonical lift of the deformation of an ordinary abelian variety

No. The picture over a general base is this: let $A_0\to S_0$ be an ordinary abelian variety over a characteristic $p$ scheme $S_0$, and let $S_n$ ($n\geq 0$) be compatible flat liftings of $S_0$ over ...
14 votes
Accepted

Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

More is true. Let $K/F$ be a field extension. Let $X$ be the vanishing set of some polynomials in $K[X_1,\dots, X_n]$. If $X$ contains a Zariski dense set of points with coordinates in $F$, then $X$ ...
  • 122k
13 votes

Adjoining torsion points from abelian varieties

Another proof that $L = \,\overline{\bf \!Q\!}\,$: Clearly $L$ is contained in $\,\overline{\bf \!Q\!}\,$, so we need only show $L$ contains every algebraic number $x \notin \bf Q$. Let $P(X)$ be the ...
13 votes
Accepted

Albanese variety over non-perfect fields

The arguments of Serre can be in fact made to work over any separably closed field. The result in the general case can then be deduced using Galois descent. Details can be found in Section 2 and the ...
13 votes

Albanese variety over non-perfect fields

The answer is affirmative (in Serre's formulation via principal homogeneous spaces) for proper, geometrically reduced, and geometrically connected schemes $X$ over any field $k$, giving a strong ...
13 votes

Bhargava's work on the BSD conjecture

For a prime $p$ and an elliptic curve $E/\mathbb{Q}$, we have the exact sequence $$\displaystyle 0 \rightarrow E(\mathbb{Q})/p E(\mathbb{Q}) \rightarrow S_p(E) \rightarrow \text{Sha}_E[p]\rightarrow ...
13 votes

Is every abelian variety a subvariety of a Jacobian?

Let me give an answer for $k = \mathbb{C}$. By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian. Now just apply Matsusaka's theorem to ...
13 votes
Accepted

Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

We give a uniform approach to $p \leq 61$ by applying analytic discriminant bounds to the Hilbert class field. To be sure this is not entirely "conceptual", but then some computation is needed even to ...
12 votes
Accepted

Adjoining torsion points from abelian varieties

If $\lambda\in\overline{\mathbb{Q}}$, the elliptic curve $$ E_\lambda\colon y^2=x(x-1)(x-\lambda) $$ has $(\lambda,0)$ as $2$-torsion point and is defined over (a subfield of) $L=\mathbb{Q}(\lambda)$....
12 votes
Accepted

Faltings height in short exact sequences

I think the following should give a counterexample. Let $\mathcal{O}$ be an order in an imaginary quadratic field $K$ and $\mathcal{O}_K$, the ring of integers. Then it's not too hard to find a (non-...
  • 2,461
12 votes
Accepted

Abelian varieties with good reduction everywhere over function fields

There are non-isotrivial families of supersingular abelian varieties of dimension $g$ over $\mathbb P^1_{\overline {\mathbb F_p}}$ if $g\geq 2$; see Goren, E. Z.(3-MGL); Oort, F. Stratifications of ...
12 votes
Accepted

Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

A conjecture of Coleman asserts that only finitely many rings arise as the endomorphism ring of an abelian variety of given dimension defined over a number field of given degree. See [1] for an ...
11 votes

Torsion points of abelian varieties in the perfect closure of a function field

The answer to question (*) is yes. It is Theorem 1.2.2 in the following preprint.
11 votes
Accepted

Shafarevich conjecture for abelian varieties

Let $B$ be a smooth projective curve over an algebraically closed field of characteristic zero. Let $K$ be the function field of $B$. Let $S$ be a finite set of closed points of $B$. You might find ...
11 votes
Accepted

Galois action on $p$-adic Tate module of Abelian variety over finite field semisimple?

You're just asking whether Frobenius acts semsimply on the $p$-adic Tate module. We know from Tate's theorem that Frobenius acts semisimply on the $\ell$-adic Tate module, and hence satisfies some ...
  • 122k
11 votes
Accepted

Tamagawa numbers

Denote by $\Phi$ the quotient of $\mathcal A^\vee$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $A^\vee$, by the connected component of $0$ of $\...
  • 9,640
11 votes
Accepted

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, ...
  • 2,461
11 votes

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 ...
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