Heegner Points and Rankin L-Series

Thirteen articles by leading contributors on the history of the Gross-Zagier formula and its developments.

Henri Darmon (Edited by), Shou-wu Zhang (Edited by)

9780521836593, Cambridge University Press

Hardback, published 21 June 2004

380 pages
24.3 x 16.1 x 2.6 cm, 0.71 kg

"The volume has an excellent array of topics and it is written by the leading mathematicians in the field. Each article serves well as an overview of the main concepts and definitely encourages the reader to pursue a deeper study of the field." MAA Reviews, Alvara Lozano-Robledo, Cornell University

The seminal formula of Gross and Zagier relating heights of Heegner points to derivatives of the associated Rankin L-series has led to many generalisations and extensions in a variety of different directions, spawning a fertile area of study that remains active to this day. This volume, based on a workshop on Special Values of Rankin L-series held at the MSRI in December 2001, is a collection of thirteen articles written by many of the leading contributors in the field, having the Gross-Zagier formula and its avatars as a common unifying theme. It serves as a valuable reference for mathematicians wishing to become further acquainted with the theory of complex multiplication, automorphic forms, the Rankin-Selberg method, arithmetic intersection theory, Iwasawa theory, and other topics related to the Gross-Zagier formula.

1. Preface Henri Darmon and Shour-Wu Zhang
2. Heegner points: the beginnings Bryan Birch
3. Correspondence Bryan Birch and Benedict Gross
4. The Gauss class number problem for imaginary quadratic fields Dorian Goldfeld
5. Heegner points and representation theory Brian Conrad (with an appendix by W. R. Mann)
6. Special value formulae for Rankin L-functions Vinayak Vatsal
7. Gross-Zagier formula for GL(2), II Shou-Wu Zhang
8. Special cycles and derivatives in Eisenstein series Stephen Kudla
9. Faltings' height and the Derivatives of Eisenstein series Tonghai Yang
10. Elliptic curves and analogies between number fields and function fields Doug Ulmer
11. Heegner points and elliptic curves of large rank over function fields Henri Darmon
12. Periods and points attached to quadratic algebras Massimo Bertolini and Peter Green.

Subject Areas: Number theory [PBH]