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Equidistribution for Tannakian monodromy groups

Beat Zurbuchen

Abstract

We prove that a perverse sheaf on a connected commutatitve algebraic group over a finite is generically unramified. This implies an equidistribution theorem for Tannakian monodromy groups in previously unavailable generality. We also prove a stratification theorem for exponential sums in families indexed by a scheme and the characters of a connected commutative algebraic group. Our method is based on Tannakian categories introduced by Gabber and Loeser. This method naturally yields fiber functors. We also prove vanishing theorems over a connected commutative algebraic group, classify the negligible sheaves, and prove relative weak propagation theorems for tori.

Equidistribution for Tannakian monodromy groups

Abstract

We prove that a perverse sheaf on a connected commutatitve algebraic group over a finite is generically unramified. This implies an equidistribution theorem for Tannakian monodromy groups in previously unavailable generality. We also prove a stratification theorem for exponential sums in families indexed by a scheme and the characters of a connected commutative algebraic group. Our method is based on Tannakian categories introduced by Gabber and Loeser. This method naturally yields fiber functors. We also prove vanishing theorems over a connected commutative algebraic group, classify the negligible sheaves, and prove relative weak propagation theorems for tori.
Paper Structure (65 sections, 156 theorems, 466 equations)

This paper contains 65 sections, 156 theorems, 466 equations.

Table of Contents

  1. Introduction
  2. Statements
  3. Stratification theorems
  4. Vanishing theorems
  5. Relative weak propagation
  6. Classification of negligible sheaves
  7. Tannakian categories
  8. Unramified characters
  9. Tannakian categories
  10. Convolution product
  11. Character theory
  12. Preliminaries
  13. Fundamental group
  14. Weil group cohomology
  15. Functoriality of the fundamental group
  16. ...and 50 more sections

Key Result

Theorem 1

Let $M \in \text{Perv}_{}({G}).$ Then $M$ is generically unramified.

Theorems & Definitions (349)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Example 1.1
  • ...and 339 more