Henri Darmon is a professor at McGill University. He is the Director of Centre Interuniversitaire en Calcul Mathematique Alegrigue and the Editor-in-chief of the Canadian Journal of Mathematics. An active researcher, he has written numerous mathematical articles and monographs. He is also the recipient of several awards such as fellow of the Royal Society of Canada, the Ribenboim Prize of the Canadian Number Theory Association and the Carrie M. Deric Award for teaching. Shou-Wu Zhang is a Professor of Mathematics at Columbia University. He has been Chair Professor at Tsinghua University and Chinese Academy of Sciences in Beijing, China. He has written articles for numerous mathematical journals, including Annals of Mathematics, Inventiones, and Journal of American Mathematics Society.

• 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.

"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