Automorphic Forms and Applications

The theory of automorphic forms has seen dramatic developments in recent years. In particular, important instances of Langlands functoriality have been established. This volume presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on automorphic forms and their applications. It addresses some of the general aspects of automorphic forms, as well as certain recent advances in the field. The book starts with the lectures of Borel on the basic theory of automorphic forms, which lay the foundation for the lectures by Cogdell and Shahidi on converse theorems and the Langlands-Shahidi method, as well as those by Clozel and Li on the Ramanujan conjectures and graphs. The analytic theory of GL(2)-forms and $L$-functions are the subject of Michel's lectures, while Terras covers arithmetic quantum chaos. The volume also includes a chapter by Vogan on isolated unitary representations, which is related to the lectures by Clozel. This volume is recommended for independent study or an advanced topics course. It is suitable for graduate students and researchers interested in automorphic forms and number theory. Information for our distributors: Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Table of Contents:
Introduction Armand Borel, Automorphic Forms on Reductive Groups: Automorphic forms on reductive groups Bibliography L. Clozel, Spectral Theory of Automorphic Forms: Spectral theory of automorphic forms Mostly SL(2) The spectral decomposition of $L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$: Arthur's conjectures Known bounds for the cuspidal spectrum and the Burger-Sarnak method Applications: Control of the spectrum All reductive adelic groups are tame Bibliography James W. Cogdell, $L$-functions and Converse Theorems for $GL_n$: $L$-functions and converse theorems for $GL_n$ Fourier expansions and multiplicity one Eulerian integrals for $GL_n$ Local $L$-functions Global $L$-functions Converse theorems Converse theorems and functoriality Bibliography Philippe Michel, Analytic Number Theory and Families of Automorphic $L$-functions: Analytic number theory and families of automorphic $L$-functions Analytic properties of individual $L$-functions A review of classical automorphic forms Large sieve inequalities The subconvexity problem Some applications of subconvexity Bibliography Freydoon Shahidi, Langlands-Shahidi Method: Langlands-Shahidi Method Basic concepts Eisenstein series and $L$-functions Functional equations and multiplicativity Holomorphy and boundedness; Applications Bibliography Audrey Terras, Arithmetical Quantum Chaos: Arithmetical quantum chaos Finite models Three symmetric spaces Bibliography David A. Vogan, Jr., Isolated Unitary Representations: Isolated unitary representations Bibliography Wen-Ching Winnie Li, Ramanujan Graphs and Ramanujan Hypergraphs: Ramanujan graphs and Ramanujan hypergraphs Ramanujan graphs and connections with number theory Ramanujan hypergraphs Bibliography.