Posted by Martin Orr on
Friday, 04 March 2016 at 11:40

Today I am going to write about Gross's proof of the Chowla-Selberg formula (up to algebraic numbers).
As I discussed last time, the Chowla-Selberg formula is a formula for the periods of a CM elliptic curve

in terms of values of the gamma function.
The idea of Gross's proof is to construct a family of abelian varieties equipped with a section of the de Rham cohomology which has constant periods, such that there is one abelian variety in the family where the period is easy to calculate, and another abelian variety in the family is a power of

and so knowing a period of this variety allows us to calculate the periods of

.

Tags
abelian-varieties, hodge, maths, number-theory, shimura-varieties
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Posted by Martin Orr on
Wednesday, 17 December 2014 at 19:00

Today I will outline the proof of Deligne's theorem that Hodge classes on an abelian variety are absolute Hodge.
The proof goes through three steps of reducing to increasingly special types of abelian varieties, until finally one reaches a case where it is easy to finish off.
This post has ended up longer than usual, but I don't think it is worth splitting into two.

A key ingredient is Deligne's Principle B, which is used for two of the three reduction steps.
Principle B says that if we have a family of varieties and a flat section of the relative de Rham cohomology bundle

, such that the section specialises to an absolute Hodge class at one point of , then in fact it is absolute Hodge everywhere.
This means that, if we have a method for constructing suitable families of abelian varieties and sections of their relative de Rham cohomology, then we only have to prove that Hodge classes are absolute Hodge at one point of each relevant family.
We use Shimura varieties to construct these families of abelian varieties on which to apply Principle B.

The outline of the proof looks like this:

- Reduce to Hodge classes on abelian varieties of CM type (using Principle B)
- Reduce to a special type of Hodge classes, called
*Weil classes*, on a special type of abelian variety, called *abelian varieties of split Weil type* (using linear algebra)
- Reduce to Hodge classes on abelian varieties which are isogenous to a power of an elliptic curve (using Principle B)
- Observe that it is easy to prove Deligne's theorem (and indeed the Hodge conjecture) for abelian varieties which are isogenous to a power of an elliptic curve

Tags
abelian-varieties, alg-geom, hodge, maths, shimura-varieties
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Posted by Martin Orr on
Thursday, 20 September 2012 at 17:39

I recently had to write an account of my research for a non-specialist. I think I took it more seriously than necessary -- I am not sure that anyone will read it. But it might interest some readers of this blog, so I shall put it here. I have just put the paper containing this work on Arxiv.

Tags
alg-geom, maths, shimura-varieties
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