Historical introduction

The p-adic theory of logarithmic forms has a long history, following closely the results in the complex domain; and it has been applied to Leopoldt's conjecture on p-adic regulators (for abelian extensions of Q, see Ax 1965 and Brumer 1967), to polynomial and exponential Diophantine equations, to the problem of the greatest prime divisors of polynomials or binary forms, to linear recurrence sequences (see Shorey & Tijdeman 1986), to knot theory (see Riley 1990) and to the abc-conjecture (see Stewart & Tijdeman 1986, Stewart & Yu 1991, 2001), etc. The present report will emphasize the evolution of the theory of p-adic logarithmic forms and its application to the abc-conjecture.

Mahler (1932) proved the p-adic analogue of the Hermite–Lindemann theorem. In 1935, he obtained a p-adic analogue of the Gel'fond–Schneider Theorem. During the course of this work, he founded the p-adic theory of analytic functions.

Gel'fond (1940) proved a quantitative result on linear forms in two p-adic logarithms in analogy with his classic work on Hilbert's seventh problem relating to two complex logarithms. Schinzel (1967) refined Gel'fond's results, giving completely explicit bounds.

At the end of his 1952 book, Gel'fond wrote ‘Nontrivial lower bounds for linear forms, with integral coefficients, of an arbitrary number of logarithms of algebraic numbers, obtained effectively by methods of the theory of transcendental numbers, will be of extraordinarily great significance in the solution of very difficult problems of modern number theory.’