A Course in p-adic Analysis

  title={A Course in p-adic Analysis},
  author={Alain M. Robert},
1 p-adic Numbers.- 2 Finite Extensions of the Field of p-adic Numbers.- 3 Construction of Universal p-adic Fields.- 4 Continuous Functions on Zp.- 5 Differentiation.- 6 Analytic Functions and Elements.- 7 Special Functions, Congruences.- Specific References for the Text.- Tables.- Basic Principles of Ultrametric Analysis.- Conventions, Notation, Terminology. 


We give a brief and elementary introduction to p-adic numbers and p-adic functions. Some of the topics are: non-archimedean valuations and the ultrametric topology, completions of Q, the Hasse

On p-adic analytic continuation with applications to generating elements

Abstract Given a prime number p and the Galois orbit O(T) of an integral transcendental element T of , the topological completion of the algebraic closure of the field of p-adic numbers, we study the

A Note on Complex p-Adic Exponential Fields

  • A. Bleybel
  • Mathematics
    p-Adic Numbers, Ultrametric Analysis and Applications
  • 2018
In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of p-adic fields in order to get some results towards algebraic independence of p-adic exponentials for almost all primes p.

Computing p-adic L-functions of totally real number fields

It is shown how these formulas can be used to compute values and representations of p-adic L-functions of totally real number fields.

On the order of vanishing of the cyclotomic p-adic L-function

For a newform for Gamma_0(N) of even weight k, we prove that its attached p-adic L-function is not identically zero on the group Z_p of the p-adic units. If p >3, we prove that the order of vanishing

On the norm of the trace functions and applications

Given a prime number p and the Galois orbit O(x) of a transcendental element x of Cp, the topological completion of the algebraic closure of the eld of p-adic numbers, we give an estimation for the

Upper triangular operators and p-adic L-functions

For a newform f for Γ0(N) of even weight k supersingular at a prime p ≥ 5, by using infinite dimensional p-adic analysis, we prove that the p-adic L-function Lp(f,α; χ) has finite order of vanishing


The purpose of this paper is to define generalized twisted q-Bernoulli numbers by using p- adic q-integrals. Furthermore, we construct a q-analogue of the p-adic generalized twisted L-functions which

Algebraic actions of discrete groups: the $p$-adic method

We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on p-adic analysis, the second makes use of