In this paper we define the p-density of a finite subset D ⊂ Nr , and show that it gives a good lower bound for the p-adic valuation of exponential sums over finite fields of characteristic p. We also give an application: when r = 1, the p-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in D. 0. Intro...

We establish a generalization of the p-adic local monodromy theorem (of André, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called “fake annuli”. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a valuation, which in this paper is restricted to be of monomial form; we defe...

Let p be a fixed odd prime number. Throughout this paper Zp, Qp, C and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = 1 p . When one talks of q-extension, q is variously considered as and in...

We compute the p-adic regulator of cyclic cubic extensions of Q with discriminant up to 1016 for 3 < p < 100, and observe the distribution of the p-adic valuation of the regulators. We find that for almost all primes the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model a conject...

We analyze properties of the 2-adic valuation of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence.

We present analytical properties of a sequence of integers related to the evaluation of a rational integral. We also discuss an algorithm for the evaluation of the 2-adic valuation of these integers that has a combinatorial interpretation.

We address the problem of the stability of the computations of resultants and subresultants of polynomials defined over complete discrete valuation rings (e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms are highly unstable on average and we explain, in many cases, how one can stabilize them without sacrifying the complexity. On the way, we completely determine the d...

Continued fractions in R have a single definition and algorithms for approximating them are well known. There also exists a well known result which states that √ m, m ∈ Q, always has a periodic continued fraction representation. In Qp, the field of p-adics, however, there are competing and non-equivalent definitions of continued fractions and no single algorithm exists which always produces a p...

Suppose K/Q is a totally real extension of degree d = [K : Q]. Let Qp denote the completion of with respect to the p-adic valuation when p is a rational prime number and let Q,, denote IR when the valuation is the usual absolute value. This latter case is thought of as corresponding to the case where the prime p is 'infinite'. Suppose : K->QP is a (Q-)linear form. We say that is p-ad...