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 \mathbb{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 \mathbb B_{\mathrm{crys}} = \mathbb B_{\mathrm{crys},p} contained in \mathbb B_{\mathrm{dR}} = \mathbb B_{\mathrm{dR},p} , namely \mathbb B_p := \mathbb B_{\mathrm{crys},p} \otimes_{\mathbb Q_p^{\mathrm{ur}}}\bar{\mathbb Q}_p \subset \mathbb B_{\mathrm{dR},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.