Linear Algebraic Groups

The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field.

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function e^z. A central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. This book includes proofs of the main basic results (theorems of Hermite-Lindemann, Gelfond-Schneider, 6 exponentials theorem), an introduction to height functions with a discussion of Lehmer's problem, several proofs of Baker's theorem as well as explicit measures of linear independence of logarithms. An original feature is that proofs make systematic use of Laurent's interpolation determinants. The most general result is the so-called Theorem of the Linear Subgroup, an effective version of which is also included. It yields new results of simultaneous approximation and of algebraic independence. 2 chapters written by D. Roy provide complete and at the same time simplified proofs of zero estimates (due to P. Philippon) on linear algebraic groups.

The invited papers collected in this volume address topics related to the research of Raman Parimala (plenary speaker at the upcoming ICM 2010). These themes focus primarily on the interplay between algebra, number theory, and algebraic geometry. The included contributions cover exciting research in areas such as field patching and a proof of the Serre's Conjecture II for function fields of complex surfaces.