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Results tagged with abelian-varieties
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user 1384
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.
14
votes
Can we count isogeny classes of abelian varieties?
Here's a related question on which there has been much work. If you actually need an answer to your question for some other reason and you follow this up, I'd be interested to see where it goes.
Let …
- 41.4k
8
votes
Explicit way to construct simple complex tori/abelian varieties of dimension at least 2
I don't really know what "some sort of algorithm" means, but here is a source of examples of simple abelian varieties. As you probably know, if $L$ is a lattice in $\mathbf{C}$ then $\mathbf{C}/L$ is …
- 41.4k
5
votes
Quotients of Tate modules
Although this question isn't really well-defined (you'd surely need to be more precise about the word "canonical" in the comment under the question) let me make two comments which hopefully put this t …
- 41.4k
4
votes
Modular forms reference
Just to add one more thing to what Pete said: the variety A_f that one normally attaches to f might have endomorphism ring bigger than an order in the coefficient field of f: for example if E is an el …
- 41.4k
3
votes
Accepted
Abelian varieties of CM type?
IIRC I learnt a lot from Katz' papers from the 1970s. Of course the basic construction is the same as the elliptic curve case: you take C^g, quotient out by the lattice coming from E via its g embeddi …
- 41.4k
3
votes
Isomorphism on p-torsion of Neron models
(a) is not going to be true in general (and as Moret-Bailly says, (b) needs some more explanation).
Here's the "moral" reason (a) isn't true. Let $E$ be an elliptic curve over the rationals. If $E[p …
- 41.4k
2
votes
Accepted
Isogeny from kernel in higher dimensional abelian varieties
This question seems a bit confused.
If $D$ is an arbitrary point in the Jacobian then one cannot construct an isogeny with kernel the subgroup generated by $D$ -- as this subgroup is typically infini …
- 41.4k