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The Batyrev-Manin conjecture for DM stacks

Ratko Darda, Takehiko Yasuda

Abstract

We define a new height function on rational points of a DM (Deligne-Mumford) stack over a number field. This generalizes a generalized discriminant of Ellenberg-Venkatesh, the height function recently introduced by Ellenberg-Satriano-Zureick-Brown (as far as DM stacks over number fields are concerned), and the quasi-toric height function on weighted projective stacks by Darda. Generalizing the Manin conjecture and the more general Batyrev-Manin conjecture, we formulate a few conjectures on the asymptotic behavior of the number of rational points of a DM stack with bounded height. To formulate the Batyrev-Manin conjecture for DM stacks, we introduce the orbifold versions of the so-called $a$- and $b$-invariants. When applied to the classifying stack of a finite group, these conjectures specialize to the Malle conjecture, except that we remove certain thin subsets from counting. More precisely, we remove breaking thin subsets, which have been studied in the case of varieties by people including Hassett, Tschinkel, Tanimoto, Lehmann and Sengupta, and can be generalized to DM stack thanks to our generalization of $a$- and $b$-invariants. The breaking thin subset enables us to reinterpret Klüners' counterexample to the Malle conjecture.

The Batyrev-Manin conjecture for DM stacks

Abstract

We define a new height function on rational points of a DM (Deligne-Mumford) stack over a number field. This generalizes a generalized discriminant of Ellenberg-Venkatesh, the height function recently introduced by Ellenberg-Satriano-Zureick-Brown (as far as DM stacks over number fields are concerned), and the quasi-toric height function on weighted projective stacks by Darda. Generalizing the Manin conjecture and the more general Batyrev-Manin conjecture, we formulate a few conjectures on the asymptotic behavior of the number of rational points of a DM stack with bounded height. To formulate the Batyrev-Manin conjecture for DM stacks, we introduce the orbifold versions of the so-called - and -invariants. When applied to the classifying stack of a finite group, these conjectures specialize to the Malle conjecture, except that we remove certain thin subsets from counting. More precisely, we remove breaking thin subsets, which have been studied in the case of varieties by people including Hassett, Tschinkel, Tanimoto, Lehmann and Sengupta, and can be generalized to DM stack thanks to our generalization of - and -invariants. The breaking thin subset enables us to reinterpret Klüners' counterexample to the Malle conjecture.
Paper Structure (20 sections, 40 theorems, 158 equations, 1 table)

This paper contains 20 sections, 40 theorems, 158 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgments
  4. Twisted sectors, residue maps and ages
  5. Twisted sectors
  6. Residue maps
  7. Ages
  8. Line bundles and stable heights
  9. Raising data and unstable heights
  10. Fano stacks; the stacky Manin conjecture
  11. Compatibility with products of Fano stacks
  12. Compatibility with coarse moduli spaces
  13. The orbifold pseudo-effective cone
  14. $a$- and $b$-invariants; the stacky Batyrev--Manin conjecture
  15. $a$- and $b$-invariants and breaking thin subsets
  16. ...and 5 more sections

Key Result

Lemma 2.7

The stack $\mathcal{J}_{0}\mathcal{X}$ is smooth and proper over $F$. Moreover, if $\mathcal{J}_{0}\mathcal{X}_{\mathcal{O}_{S}}$ is flat over $\mathcal{O}_{S}$, then it is also smooth and proper over $\mathcal{O}_{S}$.

Theorems & Definitions (161)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Stacks of twisted $0$-jets
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6: Sectors
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • proof
  • ...and 151 more