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Heegner Points and Rankin L-Series

Henri Darmon and Shou-Wu Zhang, editors
Cambridge University Press
Publication Date: 
Number of Pages: 
Mathematical Sciences Research Institute Publications
[Reviewed by
Álvaro Lozano-Robledo
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The Gross-Zagier formula relates the height of certain points on the modular curve X0(N), called Heegner points, to the derivatives of a certain L-function, the Rankin L-series. The importance of this formula manifests itself in the many and very interesting applications across number theory and beyond. This volume has the Gross-Zagier formula as theme and contains thirteen articles, based on the workshop on 'Special Values of Rankin L-Series' held at the MSRI in December 2001.

The first article is probably the most interesting. Bryan Birch, who was the first person to undertake a systematic study of Heegner points, writes a historical account of the developments which led to the formulation of Gross-Zagier: from Heegner to his own correspondence with Dick Gross. The actual letters, which are included in the volume, constitute one of those rare examples of mathematics in the making.

The rest of the volume serves two purposes. Two of the articles, written by Dick Gross, and Brian Conrad together with Russell Mann, give some background material (which may seem very little to a beginner in the field) and generalize the work of Gross-Zagier to a much more general context. The other articles look back on some of the very enlightening applications of the theory and describe several extensions (totally real fields, the function field case and higher dimensional analogues). For example, in one of the papers, Dorian Goldfeld gives a sketch of his clever proof of the Gauss class number one problem, where the Gross-Zagier formula was used to show the existence of an elliptic curve of analytic rank equal to three.

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. Some articles will be of interest to everyone in the mathematical community, such as the historical introduction or Goldfeld's article, but the book gets very technical very soon. However, each article contains a very good selection of references which will help the interested reader.

Álvaro Lozano-Robledo is visiting assistant professor of mathematics at Colby College.

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.