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Heights and quantitative arithmetic on stacky curves

Brett Nasserden, Stanley Yao Xiao

TL;DR

This work develops explicit, bottom-up height machinery for stacky curves, bridging ESZ-B heights with tractable, elementary constructions. It proves that the anti-canonical height has the Northcott property exactly when the Euler characteristic is positive, and shows a deep link between stacky Vojta conjectures and the abc-conjecture, with equivalence results in the negative Euler-characteristic regime. The paper also provides quantitative counting results for points on stacky curves, including an exact order of magnitude for a key family and a linear-programming framework to bound the Northcott threshold via abc-type phenomena. Overall, it connects stacky height theory with classical Diophantine geometry, offering explicit tools to study integral points, morphisms, and arithmetic on stacky curves. The results illuminate how stacky refinements interact with positivity, leading to precise arithmetic dichotomies governed by Euler characteristics.

Abstract

In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is non-positive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space $\mathbb{P}^1$, is equivalent to the $abc$-conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.

Heights and quantitative arithmetic on stacky curves

TL;DR

This work develops explicit, bottom-up height machinery for stacky curves, bridging ESZ-B heights with tractable, elementary constructions. It proves that the anti-canonical height has the Northcott property exactly when the Euler characteristic is positive, and shows a deep link between stacky Vojta conjectures and the abc-conjecture, with equivalence results in the negative Euler-characteristic regime. The paper also provides quantitative counting results for points on stacky curves, including an exact order of magnitude for a key family and a linear-programming framework to bound the Northcott threshold via abc-type phenomena. Overall, it connects stacky height theory with classical Diophantine geometry, offering explicit tools to study integral points, morphisms, and arithmetic on stacky curves. The results illuminate how stacky refinements interact with positivity, leading to precise arithmetic dichotomies governed by Euler characteristics.

Abstract

In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is non-positive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space , is equivalent to the -conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.

Paper Structure

This paper contains 28 sections, 45 theorems, 388 equations.

Table of Contents

  1. Introduction
  2. A further elaboration of our ideas
  3. An elementary height machine on stacky curves
  4. Properties of the anti-canonical E-S-ZB height $H_{-K_{\mathfrak{X}}}$
  5. Quantitative arithmetic on stacky curves
  6. (Stacky) Heights on $M$-curves
  7. Translation between $M$-curves and stacky curves
  8. Construction of heights
  9. Integral Points on $M$-Curves
  10. Stacky curves with coarse space ${\mathbb{P}}^1$
  11. Proof of \ref{['thm:maincomarison']}
  12. Morphisms of stacky curves
  13. Northcott property of the canonical height on stacky curves
  14. On the Northcott property of canonical and anti-canonical heights on stacky curves
  15. Minimally non-negative tuples with $n = 4$
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $\mathfrak{X}$ be the stacky curve obtained by adding three half points to ${\mathbb{P}}^1$. Then the Stacky Batyrev-Manin-Malle Conjecture is true for $H_{-K_\mathfrak{X}}$.

Theorems & Definitions (88)

  • Conjecture 1.1: Stacky Batyrev-Manin-Malle Conjecture of Ellenberg-Satriano-Zureick-Brown
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Definition 2.2: arr
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Corollary 2.5
  • ...and 78 more