Langlands functoriality conjecture and number theory. (English) Zbl 1302.11035

Shahidi, Freydoon

Kobayashi, Toshiyuki (ed.) et al., Representation theory and automorphic forms. Based on the symposium, Seoul, Korea, February 14–17, 2005. Basel: Birkhäuser (ISBN 978-0-8176-4505-2/hbk). Progress in Mathematics 255, 151-173 (2008).

Summary: We discuss several applications of the recent developments in the Langlands functoriality conjecture such as the automorphy of the symmetric powers of 2-dimensional complex representations of Galois groups of number fields, lattice point problems, Ramanujan-Selberg and Sato-Tate conjectures. We conclude by explaining how these recent developments are established.
For the entire collection see [Zbl 1124.11004].

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
22E50	Representations of Lie and linear algebraic groups over local fields
11P21	Lattice points in specified regions
22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings

Keywords:

Langlands functoriality; Ramanujan-Selberg conjectures; automorphic representations; \(L\)-functions