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Results tagged with abelian-varieties
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user 2290
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.
8
votes
Accepted
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 …
- 30.5k
9
votes
Accepted
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 …
- 30.5k
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 …
- 30.5k
1
vote
Accepted
Prime divisors on the Jacobian of a genus 2 curve over $\mathbb{F}_q$ under the $n$ map
If $\iota$ denotes the hyperelliptic involution, then the condition $n[P-\infty] = [Q-\infty]$ is equivalent to $nP+\iota(Q)$ linearly equivalent to $(n+1)\infty$. In a few pathological cases, where t …
- 30.5k
23
votes
Accepted
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 …
- 30.5k
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 …
- 30.5k
1
vote
What are sufficient and necessary conditions to be a generalized Zariski surface over a fini...
K3 surfaces are unirational if and only if they have Picard number $22$ (This was a conjecture of M. Artin, now proved by Liedtke, Inv Math 2015). Unirational means that there is a surjective rational …
- 30.5k
16
votes
Accepted
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 …
- 30.5k
2
votes
Accepted
Is the stabilizer of an irreducible subvariety of an abelian variety irreducible ?
No. Take a curve in its Jacobian and pull it back by multiplication by some n. The resulting pullback is a curve invariant by the n torsion.
- 30.5k
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 $C_{n+1} \to C_n$ …
- 30.5k
6
votes
Accepted
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 …
- 30.5k
10
votes
Accepted
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 …
- 30.5k
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 …
- 30.5k
3
votes
Serre's open image theorem for products of elliptic curves over function fields via speciali...
If $K$ is a function field over an algebraically closed field and one of your elliptic curves is constant (which does not necessarily violate your hypotheses unless the constant field is the algebraic …
- 30.5k
5
votes
Accepted
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.
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