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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.
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
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
37
votes
Accepted
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 …
- 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
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
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
4
votes
Accepted
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 …
- 30.5k
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 …
- 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
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
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
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
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
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
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