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Existence of non-abelian local constants, and their properties

Sazzad Ali Biswas

Abstract

In his Ph.D. thesis, John Tate attached the (abelian) local constants to the characters of a non-Archimedean local field of characteristic zero. Robert Langlands proved the existence theorem of a non-abelian local constant of a higher-dimensional complex local Galois representation. In 1990, Helmut Koch summarized Langlands' strategy for the existence of a non-abelian local constant (group theoretically). The Brauer induction formula plays a crucial role in Langlands' proof. Robert Boltje gives a canonical version of the Brauer induction formula. In this paper, we review Langlands' strategy using Boltje's canonical Brauer induction formula. We then review various properties of local constants, some applications, and open problems.

Existence of non-abelian local constants, and their properties

Abstract

In his Ph.D. thesis, John Tate attached the (abelian) local constants to the characters of a non-Archimedean local field of characteristic zero. Robert Langlands proved the existence theorem of a non-abelian local constant of a higher-dimensional complex local Galois representation. In 1990, Helmut Koch summarized Langlands' strategy for the existence of a non-abelian local constant (group theoretically). The Brauer induction formula plays a crucial role in Langlands' proof. Robert Boltje gives a canonical version of the Brauer induction formula. In this paper, we review Langlands' strategy using Boltje's canonical Brauer induction formula. We then review various properties of local constants, some applications, and open problems.

Paper Structure

This paper contains 21 sections, 22 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Boltje's canonical Brauer induction formula and extendible functions
  3. Brauer Canonical induction formula
  4. The Map $a_G$
  5. Extendible functions
  6. Existence proof of non-abelian local constants
  7. Local root numbers and their properties
  8. Local Fields and their finite extensions
  9. Local Constants (or local root numbers/local epsilon factors)
  10. Some properties of $W(\chi,\psi)$
  11. Connection of different conventions for local constants
  12. Local constants for virtual representations
  13. Applications, and Open Problems
  14. Applications
  15. Open Problems
  16. ...and 6 more sections

Key Result

Theorem 1.1

i. If $G$ is abelian, the kernel $Ker(b_G)$ is generated as an abelian group by the relation of type I. ii. If $G$ is nilpotent, the kernel $Ker(b_G)$ is generated as an abelian group by the relations of type I, and type II. iii. If $G$ is solvable, the kernel $Ker(b_G)$ is generated by relations

Theorems & Definitions (58)

  • Theorem 1.1: Langlands-Deligne, D1
  • Remark 1.2
  • Theorem 1.3: Langlands, Theorem 3.1 of HK
  • Remark 1.4
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4: RB2, Theorem 2.1
  • Remark 2.5
  • Remark 2.6: Weakly Extendible Functions, and Langlands $\lambda$-functions
  • ...and 48 more