We prove a vanishing theorem for the p-adic cohomology of exponential sums on An. In particular, we obtain new classes of exponential sums on An that have a single nonvanishing p-adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

We prove a version of van der Corput's Lemma for polynomials over the p-adic numbers.

In this paper, we construct a digraph structure on padic dynamical systems defined by rational functions. We study the conditions under which the functions are measure-preserving, invertible and isometric, ergodic, and minimal on invariant subsets, by means of graph theoretic properties.

The goal of this paper is to give a preliminary formalization of the p-adic numbers, in the context of the Univalent Foundations. We also provide the corresponding code verifying the construction in the proof assistant Coq. Because work in the univalent setting is ongoing, the structure and organization of the construction of the p-adic numbers we give in this paper is expected to change as Coq...

This paper addresses the question of the complexity of the decision problem for the theory Th(Q,) of p-ad ic numbers. The best known lower bound for the theory is double exponential alternating time with a linear number of alternations. I have designed an algorithm that determines the truth value of sentences of the theory requiring double exponential space. My algorithm is based on techniques ...

In the recent, many mathematicians studied the multiple zeta function in the complex number field. In this paper we construct the p-adic analogue of multiple zeta function which interpolates the generalized multiple Bernoulli numbers attached to χ at negative integers. §1. Introduction Let p be a fixed prime. Throughout this paper Z p , Q p , C and C p will, respectively, denote the ring of p-a...

Throughout this paper, let p be an odd prime number. The symbol, p, p , and p denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of p , respectively. Let be the set of natural numbers and ∪ {0}. Let νp be the normalized exponential valuation of p with |p|p p−νp p 1/p. Note that p {x | |x|p ≤ 1} lim← N /p p. ...

We study the simultaneous zeros of a random family of d polynomials in d variables over the p-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the d-fold Cartesian product of the p-adic integers. Considering models in which the maximum degree that each variable appears is N , this expected value is pdblogp Nc ( 1 + p−1 + p...

In this paper, we aim to study valuations on finite extensions of Q. These extensions fall under a special type of field called a global field. We shall also cover the topics of the Approximation Theorems and the ring of adeles, or valuation vectors.