JournalsrlmVol. 17, No. 2pp. 175–198

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

  • Francesco Baldassarri

    Università di Padova, Italy
Étale and crystalline beta and gamma functions via Fontaine's periods cover

Abstract

We compare the Ihara--Anderson theory of the pp-adic \'etale beta function, which describes the Galois action on pp-adic \'etale homology for the tower of Fermat curves over Q\mathbb{Q} of degree a power of pp, with the crystalline theory of Dwork--Coleman, based on the calculation of the Frobenius action on pp-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\mathbb{B}_{crys} = \mathbb{B}_{crys,p} contained in BdR=BdR,p\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 pp-adic \'etale gamma function of Anderson and the Morita--Dwork--Coleman pp-adic crystalline gamma function.

Cite this article

Francesco Baldassarri, Étale and crystalline beta and gamma functions via Fontaine's periods. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 2, pp. 175–198

DOI 10.4171/RLM/462