Heegner points and representation theory. (English) Zbl 1126.11032

Gross, Benedict H.

Darmon, Henri (ed.) et al., Heegner points and Rankin \(L\)-series. Papers from the workshop on special values of Rankin \(L\)-series, Berkeley, CA, USA, December 2001. Cambridge: Cambridge University Press (ISBN 0-521-83659-X/hbk; 0-511-20831-6/e-book). Mathematical Sciences Research Institute Publications 49, 37-65 (2004).

Originating in the theory of Heegner points on modular curves the author proposes conjectures which relate values of automorphic \(L\)-functions with quantities associated to Shimura varieties. Results of J. L. Waldspurger [Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)] and B. H. Gross, D. B. Zagier [Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)] should be seen as particular cases of these conjectures. Further evidence is provided by the work of S.-W. Zhang [Asian J. Math. 5, No. 2, 183–290 (2001; Zbl 1111.11030), Darmon, Henri (ed.) et al., Heegner points and Rankin \(L\)-series. Papers from the workshop on special values of Rankin \(L\)-series, Berkeley, CA, USA, December 2001. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 49, 191–241 (2004; Zbl 1126.11026)]. The author arrives at his conjectures step by step, in a way which can be easily followed by the interested reader.
For the entire collection see [Zbl 1051.11004].

Reviewer: Florin Nicolae (Berlin)

MSC:

11G18	Arithmetic aspects of modular and Shimura varieties
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

Heegner point; Shimura variety; automorphic L-function

Citations:

Zbl 0567.10021; Zbl 0608.14019; Zbl 1111.11030; Zbl 1126.11026