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. This book gives 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. The first part covers basic material in transcendental number theory but with a modern perspective. The remainder assumes some background in Lie algebras and group varieties, and covers, in some instances for the first time in book form, several advanced topics. The final chapter summarises other aspects of Diophantine geometry including hypergeometric theory and the Andre-Oort conjecture. A comprehensive bibliography rounds off this definitive survey of effective methods in Diophantine geometry.
Alan Baker ,FRS, is Emeritus Professor of Pure Mathematics in the University of Cambridge and Fellow of Trinity College, Cambridge. He has received numerous international awards, including, in 1970, a Fields medal for his work in number theory. This is his third authored book: he has edited four others for publication.
Preface. 1. Transcendence origins; 2. Logarithmic forms; 3. Diophantine problems; 4. Commutative algebraic groups; 5. Multiplicity estimates; 6. The analytic subgroup theorem; 7. The quantitative theory; 8. Further aspects of Diophantine geometry; Bibliography; Index.