The L-function and C-function of twisted T -adic exponential sums are defined. The Hodge bound for the T -adic Newton polygon of the C-function is established. As an application, the T -adic Newton polygons of the L-functions of twisted p-power order exponential sums associated to diagonal forms are explicitly given. 1. Preliminaries Let Fq be the field of characteristic p with q elements, and ...

Throughout this paper Z, Q, C, Zp, Qp and Cp will respectively denote the ring of rational integers, the field of rational numbers, the filed of complex numbers, the ring p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp such that |p|p = p −vp(p) = p. If q ∈ Cp, we normally assu...

We prove that for any field k of characteristic p > 0, any separated scheme X of finite type over k, and any overconvergent F -isocrystal E over X, the rigid cohomology H i rig(X, E) and rigid cohomology with compact supports H i c,rig(X, E) are finite dimensional vector spaces over an appropriate p-adic field. We also establish Poincaré duality and the Künneth formula with coefficients. The ar...

Let p be a prime number with p ≡ 1(mod 2). Throughout this paper, Zp,Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp. The p-adic norm | · |p is normalized as |p|p = 1 p . Let q be an indeterminate in Cp such that |1− q|p < p −1 p−1 . The q-extension of number x be defined as [x]q = 1−qx 1−q . Note that limq→1[...

This article can be read as a companion and sequel to [DLR], which proposes a conjectural expression for the so-called p-adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f , defined ov...

Throughout this paper Z,Zp,Qp and Cp will be denoted by the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp, respectively, cf. [7, 8, 9, 10]. Let νp be the normalized exponential valuation of Cp with |p|p = p −νp(p) = p. When one talks of qextension, q is variously considered as an indeterminate, a comple...

In the course of Deligne’s proof of the purity theorem, he makes certain monodromy constructions which do not a priori yield legitimate `-adic sheaves. In order to make the proof work, it therefore becomes necessary to slightly enlarge the category of sheaves considered from `-adic (constructible) sheaves to Weil sheaves. These satisfy very similar formal properties to ordinary `-adic sheaves, ...

Herrlich showed that a Mumford curve of genus g > 1 over the p-adic complex field Cp has at most 48(g− 1), 24(g− 1), 30(g− 1) or 12(g− 1) automorphisms as p = 2, 3, 5 or p > 5. The Mumford curves attaining these bounds are uniformised by normal subgroups of finite index in certain p-adic triangle groups ∆p for p ≤ 5, or in a p-adic quadrangle group p for p > 5. The finite groups attaining these...

We provide a modular method for computing the splitting field Kf of an integral polynomial f by suitable use of the byproduct of computation of its Galois group Gf by p-adic Stauduhar’s method. This method uses the knowledge of Gf with its action on the roots of f over a p-adic number field, and it reduces the computation of Kf to solving systems of linear equations modulo some powers of p and ...

We study α-adic expansions of numbers in an extension field, that is to say, left infinite representations of numbers in the positional numeration system with the base α, where α is an algebraic conjugate of a Pisot number β. Based on a result of Bertrand and Schmidt, we prove that a number belongs to Q(α) if and only if it has an eventually periodic α-expansion. Then we consider α-adic expansi...