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Torsors for finite group schemes of bounded height

Ratko Darda, Takehiko Yasuda

Abstract

Let $F$ be a global field. Let $G$ be a non trivial finite étale tame $F$-group scheme. We define height functions on the set of $G$-torsors over $F,$ which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of $G$-torsors over $F$ of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case $G$ is commutative. When $F$ is a number field, the leading constant is expressed as a product of certain arithmetic invariants of $G$ and a volume of a space attached to $G$. Moreover, an equidistribution property of $G$-torsors in the space is established.

Torsors for finite group schemes of bounded height

Abstract

Let be a global field. Let be a non trivial finite étale tame -group scheme. We define height functions on the set of -torsors over which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of -torsors over of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case is commutative. When is a number field, the leading constant is expressed as a product of certain arithmetic invariants of and a volume of a space attached to . Moreover, an equidistribution property of -torsors in the space is established.
Paper Structure (17 sections, 22 theorems, 103 equations)

This paper contains 17 sections, 22 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Malle conjecture
  3. Non-constant $G$
  4. Principal results
  5. Overview
  6. Acknowledgements
  7. Conventions
  8. Heights
  9. Cohomologies
  10. Definition of $G_*$
  11. Counting function
  12. Residue map
  13. Heights
  14. Conjecture
  15. Proof for $G$ commutative
  16. ...and 2 more sections

Key Result

Theorem 1.3.2

Conjecture introversionconj is valid for commutative $G$.

Theorems & Definitions (59)

  • Conjecture 1.1.1: Malle Malle
  • Conjecture 1.3.1
  • Theorem 1.3.2
  • Theorem 1.3.3
  • Definition 2.1.1
  • Lemma 2.2.1
  • proof
  • Definition 2.2.2
  • Remark 2.2.3
  • Lemma 2.2.4
  • ...and 49 more