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The Chowla-Selberg formula

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 André-Oort conjecture for \mathcal{A}_g.

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.

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