Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stacky Batyrev-Manin conjecture and modular curves

Ratko Darda, Changho Han

Abstract

Let $\mathscr{X}_0(N)$ be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational $N$-isogeny over a number field $F$. Let $N\in\{1,2,3,4,5,6,7,8,9,10,12,13,16,18,25\},$ which are precisely the values for which the coarse moduli space of $\mathscr{X}_0(N)$ is isomorphic to $\mathbb{P}^1$. We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on $\mathscr{X}_0(N)$ when $F=\mathbb{Q}$. In the process, we give a concrete description of $\mathscr{X}_0(N)$ as a square root stack over a stacky curve.

Stacky Batyrev-Manin conjecture and modular curves

Abstract

Let be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational -isogeny over a number field . Let which are precisely the values for which the coarse moduli space of is isomorphic to . We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on when . In the process, we give a concrete description of as a square root stack over a stacky curve.
Paper Structure (24 sections, 55 theorems, 155 equations)

This paper contains 24 sections, 55 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Batyrev--Manin conjecture and stacks
  3. Results
  4. Strategy of proof of Theorem \ref{['valuesaandb']}
  5. Organization
  6. Notation
  7. Acknowledgments
  8. Preliminaries
  9. Cyclotomic stacks and the stacky Proj construction
  10. Root stacks
  11. Sectors of some Deligne--Mumford curves
  12. Weighted projective stacks
  13. Modular curves
  14. Naive height
  15. Canonical element
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $N \in \mathfrak{N}_0$ as above and let $M_{\mathop{\mathrm{naive}}\nolimits}\in\mathop{\mathrm{NS}}\nolimits_{\mathop{\mathrm{orb}}\nolimits}(\mathscr X_0(N))$ be the element corresponding to the naive height $H$. Assume that $F=\mathbb Q$. Then Conjecture mainconjecture holds for $\mathscr X_0

Theorems & Definitions (121)

  • Conjecture 1.1: dardayasudabm
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['thm:mainthm']} assuming Theorem \ref{['valuesaandb']}
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Example 2.1
  • Lemma 2.2
  • ...and 111 more