Theory of automorphic forms of weight 1. (English) Zbl 0658.10031

Investigations in number theory, Adv. Stud. Pure Math. 13, 503-584 (1988).
[For the entire collection see Zbl 0646.00003.]
The author describes a number of recent results in the theory of weight one automorphic forms, and applications to number theory. Included in this work is a very readable account of various aspects of the theory including Langland’s conjecture, the theory of complex multiplication, reciprocity laws and relations with Fourier coefficients, Galois representations, the Deligne-Serre theorem, Stark’s conjecture, and traces of Hecke operators acting on the space of weight one modular forms. The paper contains a number of beautiful examples and results. For example, the author produces a number-theoretic derivation of results of Kac and Peterson which were originally obtained using the theory of string functions for infinite dimensional affine Lie algebras.
Reviewer: S.Kamienny


11F12 Automorphic forms, one variable
11G15 Complex multiplication and moduli of abelian varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11R70 \(K\)-theory of global fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols


Zbl 0646.00003