Étale and crystalline beta and gamma functions via Fontaine's periods

Abstract

We compare the Ihara-Anderson theory of the p-adic Étale beta function, which describes the Galois action on p-adic Étale homology for the tower of Fermat curves over Q of degree a power of p, with the crystalline theory of Dwork-Coleman, based on the calculation of the Frobenius action on p-adic de Rham cohomology of the same curves. The two constructions are easily related via a ramified extension of Fontaine's period ring Bcrys = Bcrys, p contained in BdR = BdR,p, namely B p:= Bcrys, p×Qurp Q̄p ⊂ BdR, p. We propose, but do not carry out, a similar comparison for the p-adic Étale gamma function of Anderson and the Morita-Dwork-Coleman p-adic crystalline gamma function.