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

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 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 . 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.