The Frobenius and monodromy operators for curves and abelian varieties

Abstract

In this paper we will give explicit descriptions of Hyodo and Kato's Frobenius and Monodromy operators on the first p-adic de Rham cohomology groups of curves and Abelian varieties with semi-stable reduction over local fields of mixed characteristic. This paper was motivated by the first author's paper [C-pSI], where conjectural definitions of these operators for curves with semi-stable reduction were given. Although B.LeStum wrote a paper entitled "La structure de Hyodo-Kato pour les courbes" ([LS]) he did not prove or claim to have proved that the operators he defined there were the same as Hyodo and Kato's.The paper is naturally divided into two chapters. In Chapter I, written by the

Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $L$-functions

“…The cohomology of semistable curves. This section recalls the description (following [CI99], and [CI10]) of the de Rham cohomology of a semistable curve and the attendant structures with which it is equipped. Let F be a finite extension of Q p , let O F denote its ring of integers, fix an uniformizer π F and let k F = O F /π F denote the residue field.…”

New methods for $$(\varphi, \Gamma)$$ ( φ , Γ ) -modules

“…6 We describe here a common variant of W r− (F ). given by Fargues and Fontaine [8]; this interpretation is facilitated by some useful noetherian properties of these rings, as described in [17].…”

“…In the process, we obtain a far more general result, in which Q p (µ p ∞ ) can be replaced by any sufficiently ramified p-adic field; more precisely, we obtain a functorial (and hence compatible with Galois theory) correspondence between perfectoid fields and perfect fields of characteristic p. This is the tilting correspondence in the sense of [21], which is proved using almost ring theory; our proof here is the somewhat more elementary argument found in [18] (see Remark 1.5.7 for further discussion). As a historical note, we remark that we learned the key ideas from this proof from attending Coleman's 1997 Berkeley course "Fontaine's theory of the mysterious functor, " whose principal content appears in [6].…”

Abelian Surfaces With Supersingular Good Reduction and Non-Semisimple Tate Module

“…The functor V → D * cris (V ) establishes an anti-equivalence between the category of crystalline p-adic representations of G and MF ad Qp (ϕ), a quasi-inverse being V * cris (D, Fil) = Hom ϕ,Fil (D, B cris ) ( [3]). These categories are well-suited to our problem since for an abelian variety A over Q p the G-module V p (A) is crystalline if and only if A has good reduction ( [2,Theorem 4.7] …”