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Thin sets in weighted projective stacks

Stephanie Chan, Daniel Loughran, Nick Rome

TL;DR

We address counting rational points on weighted projective stacks outside thin subsets by proving a stacky Serre-type bound, driven by a new lopsided large sieve over number fields. The main technical advance is a large sieve tailored to unit-group dynamics and height on stacks, enabling sharp bounds for points avoiding prescribed residue classes modulo primes. This leads to a strong quantitative statement: 100% of odd hyperelliptic curves over a number field, ordered by weighted height, do not admit a prescribed nontrivial level structure. The results unify stack-theoretic Hilbert property considerations with moduli-space arithmetic and open avenues for counting problems on stacks beyond classical schemes.

Abstract

We prove an upper bound for the number of rational points of bounded height in a weighted projective stack which lie in a given thin subset. As a consequence, we show that $100\%$ of hyperelliptic curves do not admit a prescribed on-trivial level structure.

Thin sets in weighted projective stacks

TL;DR

We address counting rational points on weighted projective stacks outside thin subsets by proving a stacky Serre-type bound, driven by a new lopsided large sieve over number fields. The main technical advance is a large sieve tailored to unit-group dynamics and height on stacks, enabling sharp bounds for points avoiding prescribed residue classes modulo primes. This leads to a strong quantitative statement: 100% of odd hyperelliptic curves over a number field, ordered by weighted height, do not admit a prescribed nontrivial level structure. The results unify stack-theoretic Hilbert property considerations with moduli-space arithmetic and open avenues for counting problems on stacks beyond classical schemes.

Abstract

We prove an upper bound for the number of rational points of bounded height in a weighted projective stack which lie in a given thin subset. As a consequence, we show that of hyperelliptic curves do not admit a prescribed on-trivial level structure.
Paper Structure (15 sections, 11 theorems, 39 equations)

This paper contains 15 sections, 11 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Thin sets
  3. Serre's question for weighted projective space
  4. Application to level structures on hyperelliptic curves
  5. Conventions
  6. Structure of the paper
  7. Stacks and thin sets
  8. Weighted projective stacks
  9. Hilbert property for stacks
  10. A lopsided large sieve over number fields
  11. Thin sets in weighted projective stacks
  12. Level structures on hyperelliptic curves
  13. Moduli stacks of abelian varieties
  14. Moduli stacks of hyperelliptic curves
  15. Proof of Theorem \ref{['thm:level_structures']}

Key Result

Theorem 1.1

Let $\mathbf{a} = (a_0, \dots, a_n) \in \mathbb{N}^{n+1}$, let $k/\mathbb{Q}$ be a number field of degree $d$ and let $H_\mathbf{a}$ denote the weighted height function (see Definition def:height). For any thin subset $\Omega \subset \mathbb{P}(\mathbf{a})(k)$ there exists $\gamma < 1$ such that

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • proof : Proof of Theorem \ref{['thm:HP']}
  • ...and 17 more