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.