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
Read more...

Posted by Martin Orr on
Monday, 28 December 2015 at 20:00

The Chowla-Selberg formula is an equation which expresses the periods of CM elliptic curves in terms of values of the gamma function at rational arguments.
Colmez conjectured a generalisation of the Chowla-Selberg formula to higher-dimensional CM abelian varieties, and an averaged version of Colmez's conjecture was recently proved by Andreatta, Goren, Howard and Madapusi Pera and independently by Yuan and Zhang.
This has been much talked about in the world of abelian varieties, because Tsimerman used the averaged Colmez conjecture to complete the proof of the AndrĂ©-Oort conjecture for

.

I thought that rather than looking directly at the Colmez conjecture, it would be good to start with the simpler Chowla-Selberg formula i.e. the elliptic curves case.
In this post I will talk about a couple of ways of stating the formula, in terms of periods of CM elliptic curves or in terms of Faltings heights.

Tags
abelian-varieties, faltings, maths, number-theory
Read more...

Posted by Martin Orr on
Tuesday, 03 November 2015 at 16:00

In my last post, I discussed how the existence of a polarisation implies an upper bound for the transcendence degree of the extended period matrix of an abelian variety, namely the dimension of the general symplectic group

(where

is the dimension of the abelian variety).
In this post, I will discuss how this can be generalised to take into account all algebraic cycles on the abelian variety.
The group

is replaced by the motivic Galois group of the abelian variety, which I will define.
I will also mention how Deligne's theorem on absolute Hodge cycles allows us to replace the motivic Galois group by the Mumford-Tate group.

Tags
abelian-varieties, alg-geom, hodge, maths, number-theory
Read more...

Posted by Martin Orr on
Monday, 26 October 2015 at 11:00

The Legendre period relation is a classical equation relating the periods and quasi-periods of an elliptic curve, as defined last time.
I will discuss this relation, and then more generally discuss how the existence of polarisations implies relations between the periods of higher-dimensional abelian varieties.

These examples motivate the introduction of the geometric motivic Galois group, which gives an upper bound for the transcendence degree of periods of an abelian variety (or indeed any algebraic variety).
This upper bound is conjectured to be equal to the actual transcendence degree.
I had intended to discuss the geometric motivic Galois group in this post too, but I decided that it was getting to long so I will postpone that to another time.

Tags
abelian-varieties, alg-geom, hodge, maths, number-theory
Read more...