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

Abstract

We compare the Ihara--Anderson theory of the $p$-adic \'etale beta function, which describes the Galois action on $p$-adic \'etale 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 $\mathbb{B}_{crys} = \mathbb{B}_{crys,p}$ contained in $\mathbb{B}_{dR} = \mathbb{B}_{dR,p}$, namely \mathbb{B}_p := \mathbb{B}_{crys,p} \otimes_{\mathbb{Q}_p^{ur}}\bar\mathbb{Q}_p \subset \mathbb{B}_{dR,p}. We propose, but do not carry out, a similar comparison for the $p$-adic \'etale gamma function of Anderson and the Morita--Dwork--Coleman $p$-adic crystalline gamma function.