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The Weyl law for algebraic tori

Ian Petrow

Abstract

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus $T$ with bounded analytic conductor. The analytic conductor which we use is defined via the local Langlands correspondence for tori by choosing a finite dimensional complex algebraic representation of the $L$-group of $T$. Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor.

The Weyl law for algebraic tori

Abstract

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus with bounded analytic conductor. The analytic conductor which we use is defined via the local Langlands correspondence for tori by choosing a finite dimensional complex algebraic representation of the -group of . Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor.

Paper Structure

This paper contains 36 sections, 61 theorems, 448 equations.

Table of Contents

  1. Introduction
  2. Motivation and statement of results
  3. Examples
  4. Relation to the Manin conjecture
  5. Outline of the proof
  6. Index of notation
  7. Acknowledgements
  8. Background and notation
  9. Tori and groups of multiplicative type over a field
  10. $L$-groups of tori and representations
  11. Local Langlands correspondence
  12. Some tools
  13. Local non-archimedean theory
  14. Local Langlands correspondence, local conductors
  15. The Artin conductor
  16. ...and 21 more sections

Key Result

Theorem 1.1

Suppose that $r \vert_{\widehat{T}}$ is faithful. Then there exists a non-zero polynomial $P = P_{T,r,\nu}$ and $c=c_{T,r}>0$ such that If the Artin conjecture holds for the finitely many Artin $L$-functions specified in Theorem , then the error term in eq:MT may be improved to $O_{T,r,\nu}(X^{A-\delta})$ for some $\delta=\delta_{T,r}>0$. If $r \vert_{\widehat{T}}$ is not faithful, then the left

Theorems & Definitions (128)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Example 1.8
  • Conjecture 1.9: Batyrev-Manin Conj. C$'$
  • ...and 118 more