This book has arisen from lectures given by the first author at ETH Zürich in the Wintersemester 1988–1989 under the Nachdiplomvorlesung program and subsequent lectures by both authors in various localities, in particular at an instructional conference organised by the DMV in Blaubeuren. Our object has been to give an account of the theory of linear forms in the logarithms of algebraic numbers with special emphasis on the important developments of the past twenty-five years concerning multiplicity estimates on group varieties.

As will be clear from the text there is now much interplay between studies on logarithmic forms and deep aspects of arithmetic algebraic geometry. New light has been shed for instance on the famous conjectures of Tate and Shafarevich relating to abelian varieties and the associated celebrated discoveries of Faltings establishing the Mordell conjecture. We give a connected exposition reflecting these major advances including the first version in book form of the basic works of Masser and Wüstholz on zero estimates on group varieties, the analytic subgroup theorem and their applications. Our discussion here is more algebraic in character than the original and involves, in particular, Hilbert functions in degree theory and Poincaré series as well as the general background of Lie algebras and group varieties.