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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 geometry and number theory.

37 votes
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Is there any rational curve on an Abelian variety?

There are no rational curves in an abelian variety, this is much stronger than not being uniruled. If there is a map $P^1 \to A$, $A$ abelian, the map would factor through the Albanese variety of $P^1 …
Felipe Voloch's user avatar
23 votes
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On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main r...

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 pr …
Felipe Voloch's user avatar
16 votes
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Modularity theorem for abelian varieties

Abelian varieties over the rationals are modular if and only if they are of "$GL_2$"-type, which is a notion introduced by Ribet who proved that this statement is a consequence of Serre's conjecture w …
Felipe Voloch's user avatar
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 $\pro …
Felipe Voloch's user avatar
10 votes

Over which fields does the Mordell-Weil theorem hold?

Here is an [INCOMPLETE, POSSIBLY INCORRECT] answer to question 1. Yes. Let $C_n/k,n=1,2,\ldots$ be a sequence of curves of increasing genus defined over a finite field $k$ with maps $C_{n+1} \to C_n$ …
Felipe Voloch's user avatar
10 votes
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Mordell-Weil group of the universal abelian scheme

For $g=1$ there is a classical paper of Shioda (one of the two cited below) that proves that in char. zero, the group is what you expect but in char. p there are situations in which you get sections o …
Felipe Voloch's user avatar
9 votes

Torsion of an abelian variety under reduction.

If A is an abelian variety over a local field of char zero with good reduction, then any point in the kernel of reduction is in the so-called formal group. Now, to answer your question (properly under …
Felipe Voloch's user avatar
9 votes
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Is there a separable isogeny between any two isogenous abelian varieties?

The answer is no. I think Asvin's example in the comments is correct but there may be a few things to check. I will give a different example that is easier to check, using Moret-Bailly's famous exampl …
Felipe Voloch's user avatar
8 votes

Schottky locus in genus 2

By (a possible) definition, a principal polarization on an abelian surface is a curve with self-intersection 2. So, if smooth, it is a genus two curve and the abelian surface is a jacobian. You have t …
Felipe Voloch's user avatar
8 votes
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Explicit equations for the universal vector extension of an elliptic curve

(Not a complete answer but too long for a comment) My guess is that such a description is not known, probably because there isn't an easy one. There is a very nice complex analytic description as $\ma …
Felipe Voloch's user avatar
6 votes
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Serre-Tate canonical lifts for finite fields

As you just said, the canonical lift is an abelian scheme over the ring of Witt vectors $W(k)$. Now, if $k$ is finite of characteristic $p$, $W(k)$ is the ring of integers of the unramified extension …
Felipe Voloch's user avatar
5 votes

For a line bundle L on a smooth projective variety X, what is meant by Pic^L(X)

I'd guess it means the set of line bundles algebraically equivalent to L, modulo linear equivalence. In that case, for L = O_X you get Pic^0 and, in general, Pic^L is a principal homogeneous space for …
Felipe Voloch's user avatar
5 votes
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projective subvarieties of the moduli space of abelian varieties

The dimension is $g(g+1)/2$. The supersingular locus gives a large projective subvariety but I don't recall whether it is smooth or not. For references, look up the many papers of F. Oort.
Felipe Voloch's user avatar
5 votes

rank of Jacobian of Fermat curve and Chabauty-Coleman method

The rank can be estimated and Chabauty's method applied assuming a conjecture about cyclotomic fields which is still open: McCallum, William G. On the method of Coleman and Chabauty. Math. Ann. 299 …
Felipe Voloch's user avatar
4 votes
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Brauer-Manin obstruction and Tate-Shafarevich group of an Abelian variety

The quotient of what you called the Brauer-Manin obstruction by the closure of $A(K)$ within it is related to the divisible part of Sha. In particular, if Sha has no divisible part (e.g. if it is fini …
Felipe Voloch's user avatar

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