On Colmez's product formula for periods of CM-abelian varieties

TL;DR

This paper completes the proof of Colmez's conjecture on the product formula for periods of CM-abelian varieties by resolving the remaining ambiguity involving powers of 2, using Galois actions on Fermat curves.

Contribution

It finalizes Colmez's proof of the product formula for CM-abelian varieties by removing the unresolved power of 2 factor, employing analysis of Galois actions on Fermat curves in mixed characteristic.

Findings

Proof of Colmez's conjecture for K/Q abelian completed.

Analysis of Galois action on De Rham cohomology of Fermat curves.

Understanding stable reduction of Z/2^n-covers of the projective line.

Abstract

Colmez conjectured a product formula for periods of abelian varieties with complex multiplication by a field K, analogous to the standard product formula in algebraic number theory. He proved this conjecture up to a rational power of 2 for K/Q abelian. In this paper, we complete the proof of Colmez for K/Q abelian by eliminating this power of 2. Our proof relies on analyzing the Galois action on the De Rham cohomology of Fermat curves in mixed characteristic (0, 2), which in turn relies on understanding the stable reduction of Z/2^n-covers of the projective line, branched at three points.