Description
Book SynopsisIn his 1974 seminal paper 'Elliptic modules', V G Drinfeld introduced objects into the arithmetic geometry of global function fields which are nowadays known as 'Drinfeld Modules'. They have many beautiful analogies with elliptic curves and abelian varieties. They study of their moduli spaces leads amongst others to explicit class field theory, Jacquet-Langlands theory, and a proof of the Shimura-Taniyama-Weil conjecture for global function fields.This book constitutes a carefully written instructional course of 12 lectures on these subjects, including many recent novel insights and examples. The instructional part is complemented by research papers centering around class field theory, modular forms and Heegner points in the theory of global function fields.The book will be indispensable for everyone who wants a clear view of Drinfeld's original work, and wants to be informed about the present state of research in the theory of arithmetic geometry over function fields.
Table of ContentsIntroduction to Drinfeld modules and their moduli schemes, H. Matzat, M. Saidi; Analytic theory of Drinfeld modules, B. Lopez; Drinfeld modules of rank one, I. Rust; Drinfeld modules over finite fields, B. Snyder; Some rigid geometry, G. van Steen; The Bruhat-Tits building of PG(r,K infinity) and analytic moduli for Drinfeld modules, M. van der Put; Analytic compactification and modular forms, M. van der Put; Algebraic compactification and modular interpretation, J. Top; Drinfeld modular forms, G. Cornelissen; Jacquet-Langlands theory over K and relations with elliptic curves, E.U. Gekeler, M. Reversat; Some results on Hilbert class field towers of global fields, B. Angles; The Tate conjecture for elliptic surfaces over finite fields, M.L. Brown; Drinfeld modular forms of weight one, G. Cornelissen; Explicit reciprocity law for F2[t] as a Carlitz module, Y. Hellegouarch, V. Mauduit; Groups of units and torsion points of Drinfeld modules, H. Oukhaba; L-series of Drinfeld automorphic functions and quadratic field extensions, H.G. Ruck, U. Tipp; Heegner points on Drinfeld modular curves, H.G. Ruck, U. Tipp; CM-cycles in families of products E x E over finite fields, J. Top; Values and derivatives of L-series attached to elliptic curves over function fields, D. Ulmer; Norms and traces of singular moduli, D. Zagier; supersingular moduli, orthogonal polynomials and hypergeometric functions, D. Zagier.
