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Results tagged with abelian-varieties
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user 1149
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.
3
votes
Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
Theorem 6 of this paper gives a generalization of Shafarevich's Theorem:
Theorem: Let $K$ be a Hilbertian field. Let $A_{/K}$ be a nontrivial abelian variety. If $n > 1$ is indivisible by the …
- 65.4k
5
votes
Accepted
Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication
I presume I am making a very basic error somewhere but I don't see where.
Yes: you are confusing the automorphism group $\operatorname{Aut} A = (\operatorname{End} A)^{\times}$ with the automorph …
- 65.4k
8
votes
Accepted
Fourier-Mukai transform for abelian varieties
No. As Will Sawin indicates, every finite subgroup $H$ of $A$ is contained in $K(L)$ for some very ample line bundle on $A$: indeed, let $L_1$ be your favorite very ample line bundle on $A$, and let $ …
- 65.4k
9
votes
Accepted
Simple abelian varieties over non algebraically closed fields.
You're right: "simple abelian variety $A/K$" is ambiguous when $K$ is not algebraically closed. One should say geometrically (or absolutely) simple or K-simple to emphasize which is meant.
If you do …
- 65.4k
4
votes
Over which fields does the Mordell-Weil theorem hold?
This is an attempt at a relatively mild generalization of what others have said:
Let $K$ be a field and $|\cdot|: K \rightarrow \mathbb{R}$ be a nontrivial absolute value on $K$.
$\bullet$ If $K$ i …
- 65.4k
3
votes
Global Sections of the Identity Component of Neron model
I will have to wait for a more reasonable hour to give a complete answer, but I believe this paper of mine -- joint with X. Xarles -- is relevant to your question. Most of it works in the case of an …
- 65.4k
8
votes
The $Pic^0$ of an abelian variety
If I'm not mistaken, you've copied down Mumford's definition incorrectly: it should be
the set of all line bundles $L$ such that $T_x^* L \cong L$ for all $x \in A$.
Once you make this correction: …
- 65.4k
7
votes
Torsion points in Abelian varieties over number fields
Let $l$ be the field cut out by the action of the Galois group on all the torsion points of $A$, and let $\mathfrak{g} = \operatorname{Gal}(l/k)$. Then $\mathfrak{g}$ is a closed subgroup of $\operato …
- 65.4k
15
votes
Accepted
Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM?
Yes. This follows from the main result of the following paper of Zarhin.
MR0885780 (88h:14046)
Zarhin, Yu. G.
Endomorphisms and torsion of abelian varieties.
Duke Math. J. 54 (1987), no. 1, 131–145. …
- 65.4k
1
vote
possible CM-types of abelian varieties
I posted the following answer yesterday after only a quick skim of the question. When I read it with more care, it seemed to me to be the answer to a different question entirely. After having looked …
- 65.4k
4
votes
Accepted
isogenies between abelian varieties that induce isomorphisms?
Kevin's comment is right on the money, but here it is in more detail: I will give a general criterion for an isogeny $\varphi: A \rightarrow B$ of abelian varieties to induce an isomorphism upon passa …
- 65.4k
7
votes
Accepted
Is there an intrinsic way to define the group law on Abelian varieties?
Any torsor $V$ under an abelian variety over any field $K$ is caonically isomorphic to its degree $1$ Albanese variety $\operatorname{Alb}^1(V)$, which is itself a torsor under the degree $0$ Albanese …
- 65.4k
4
votes
Accepted
For a line bundle L on a smooth projective variety X, what is meant by Pic^L(X)
Let me address the last part of your question.
Let $X$ be a smooth, projective variety over an arbitrary ground field $k$.
I want to write $Pic^{[L]}(X)$ instead of $Pic^L(X)$ -- i.e., to make explici …
- 65.4k
7
votes
reduction of CM elliptic curves
A nice modern perspective on this is to use the Honda-Tate classification of endomorphism algebras of abelian varieties over a finite field. This allows you to recover the result in question (which, …
- 65.4k
15
votes
Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
My next door neighbor (in the math department) is a Hodge theorist, and I have never heard her say that abelian varieties are the hardest case of the Hodge conjecture.
However, they are certainly a de …
- 65.4k