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Counting torsors for wild abelian groups

Ratko Darda, Takehiko Yasuda

TL;DR

The paper develops a framework to count $G$-torsors over global function fields for wild abelian $p$-groups $G$ by pairing heights on wild Deligne–Mumford stacks with adelic methods. Central to the approach is the moduli stack $\Delta_G$ of formal $G$-torsors, described via Witt vectors and explicit functoriality, and the discriminant loci encoded by long flags and their irreducible strata. Under suitable (and strongly suitable) height functions, the authors prove asymptotics for the counting problem, with the discriminant height providing sharper bounds than previous tame results, and they exhibit a broad class of height functions (including representations) that yield a rich spectrum of $a(H)$ and $b(H)$ invariants. The results connect conductor/discriminant phenomena, adelic volume computations, and geometric-heuristic predictions in a coherent, quantifiable framework, enabling Northcott-type finiteness and asymptotic formulas for wild abelian torsors. This advances understanding of Malle-type distributions in positive characteristic and highlights the role of $ abla$-like height data in counting torsors over global fields.

Abstract

Let $F$ be a global field of characteristic $p > 0$ and $G$ a finite abelian $p$-group. In this paper we treat the question of counting $G$-torsors over $F$ for certain heights developed in [DY25].

Counting torsors for wild abelian groups

TL;DR

The paper develops a framework to count -torsors over global function fields for wild abelian -groups by pairing heights on wild Deligne–Mumford stacks with adelic methods. Central to the approach is the moduli stack of formal -torsors, described via Witt vectors and explicit functoriality, and the discriminant loci encoded by long flags and their irreducible strata. Under suitable (and strongly suitable) height functions, the authors prove asymptotics for the counting problem, with the discriminant height providing sharper bounds than previous tame results, and they exhibit a broad class of height functions (including representations) that yield a rich spectrum of and invariants. The results connect conductor/discriminant phenomena, adelic volume computations, and geometric-heuristic predictions in a coherent, quantifiable framework, enabling Northcott-type finiteness and asymptotic formulas for wild abelian torsors. This advances understanding of Malle-type distributions in positive characteristic and highlights the role of -like height data in counting torsors over global fields.

Abstract

Let be a global field of characteristic and a finite abelian -group. In this paper we treat the question of counting -torsors over for certain heights developed in [DY25].
Paper Structure (11 sections, 50 theorems, 253 equations)

This paper contains 11 sections, 50 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Moduli stack of formal $G$-torsors
  3. Witt vectors
  4. The stack $\Delta_G$
  5. Discriminant loci
  6. Long flags
  7. Irreducible components
  8. Commutative case
  9. Heights
  10. Appendix: a counting result
  11. Appendix: A result on commutative group stacks

Key Result

Theorem 1.2.0.1

Let $H:BG\langle F\rangle\to\mathbb R_{>0}$ be a height satisfying the conditions stated in Definition suitable. For any $\varepsilon>0$, we have that

Theorems & Definitions (129)

  • Theorem 1.2.0.1: Corollary \ref{['ainvariantcorrect']}
  • Theorem 1.2.0.2: Theorem \ref{['strsuitthm']}
  • Definition 2.0.1.1
  • Lemma 2.0.1.2
  • proof
  • Definition 2.1.1.1
  • Example 2.1.1.2
  • Lemma 2.1.1.3
  • proof
  • Lemma 2.1.1.4
  • ...and 119 more