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

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

2011-07-04

arxiv.org/abs/1107.0684v1

Mathematics

Number Theory

13 pages, comments welcome

Scientific paper

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.

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