An Introduction to Non-Abelian Class Field Theory

Automorphic Forms of Weight 1 and 2-Dimensional Galois Representations

This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.

Contents:

• Part I:

  • Higher Reciprocity Laws
  • Hilbert Class Fields over Imaginary Quadratic Fields
  • Indefinite Modular Forms
  • Dimension Formulas in the Case of Weight 1

• Part II:

  • 2-Dimensional Galois Representations of Odd Type and Non-Dihedral Cusp Forms of Weight 1
  • Maass Cusp Forms of Eigenvalue 1/4
  • Selberg's Eigenvalue Conjecture and the Ramanujan–Petersson Conjecture
  • Indefinite Theta Series
  • Hilbert Modular Forms of Weight 1
  • Appendix: Some Dimension Formula and Traces of Hecke Operators for Cusp Forms of Weight 1 — Göttingen talk, 1989. By Toyokazu Hiramatsu

Readership: Advanced undergraduate and graduate students, and researchers in number theory.
Key Features:

• The monograph provides a comprehensive overview of the author's works which include new contributions, such as dimension formulas in the case of weight 1 forms, some relations between higher reciprocity laws of non-abelian polynomials and Langlands' reciprocity laws