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Finiteness properties for Shimura curves and modified diagonal cycles

Congling Qiu

TL;DR

The paper establishes finiteness and computability results for Shimura curves with bounded gonality and for auto-critical Shimura curves, tying these properties to quaternion-algebra data, Atkin–Lehner theory, and automorphic representations. It provides a concrete, algorithmic framework—grounded in the LMFDB modular-form database and Sage—for enumerating and verifying auto-critical instances, both over ${\mathbb{C}}$ and over ${\mathbb{Q}}$ with various level structures. The main contributions include explicit finiteness theorems, a complete classification of auto-critical curves in the classical modular setting, and extensive, computable classifications for Shimura curves over ${\mathbb{Q}}$ with general levels, along with concrete examples of curves with vanishing modified diagonal cycles. These results advance the understanding of how geometric finiteness, automorphic representation theory, and explicit arithmetic data interact to constrain the landscape of critical Shimura curves and related Diagonal-cycle phenomena.

Abstract

We prove that only finitely many Shimura curves can have gonality bounded by a given number, and we study the computability of this finite set. Motivated by the relation between hyperellipticity (that is, gonality 2) and the vanishing of the modified diagonal cycle, we conjecture that such vanishing occurs for only finitely many Shimura curves. We establish several finiteness and classification results toward this conjecture and, as a by-product, obtain explicit examples of curves with vanishing modified diagonal cycles. Our computations are based on modular form data from the database \textsf{LMFDB}, and some of them are carried out using the computer algebra system \textsf{Sage}.

Finiteness properties for Shimura curves and modified diagonal cycles

TL;DR

The paper establishes finiteness and computability results for Shimura curves with bounded gonality and for auto-critical Shimura curves, tying these properties to quaternion-algebra data, Atkin–Lehner theory, and automorphic representations. It provides a concrete, algorithmic framework—grounded in the LMFDB modular-form database and Sage—for enumerating and verifying auto-critical instances, both over and over with various level structures. The main contributions include explicit finiteness theorems, a complete classification of auto-critical curves in the classical modular setting, and extensive, computable classifications for Shimura curves over with general levels, along with concrete examples of curves with vanishing modified diagonal cycles. These results advance the understanding of how geometric finiteness, automorphic representation theory, and explicit arithmetic data interact to constrain the landscape of critical Shimura curves and related Diagonal-cycle phenomena.

Abstract

We prove that only finitely many Shimura curves can have gonality bounded by a given number, and we study the computability of this finite set. Motivated by the relation between hyperellipticity (that is, gonality 2) and the vanishing of the modified diagonal cycle, we conjecture that such vanishing occurs for only finitely many Shimura curves. We establish several finiteness and classification results toward this conjecture and, as a by-product, obtain explicit examples of curves with vanishing modified diagonal cycles. Our computations are based on modular form data from the database \textsf{LMFDB}, and some of them are carried out using the computer algebra system \textsf{Sage}.
Paper Structure (18 sections, 68 theorems, 126 equations)

This paper contains 18 sections, 68 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Finitenss of Shimura curves with bounded gonality
  3. Quaternion algebras
  4. Shimura curves over ${\mathbb {C}}$
  5. Finitenss of auto-critical Shimura curves
  6. Shimura curves over a totally real number field
  7. Atkin--Lehner quotients
  8. Auto-critical curves
  9. Classification of auto-critical Shimura curves over ${\mathbb {Q}}$
  10. Algorithms and examples
  11. Classical modular curves
  12. Shimura curves over ${\mathbb {Q}}$ (I)
  13. Shimura curves over ${\mathbb {Q}}$ (II)
  14. More general levels
  15. Trilinear forms
  16. ...and 3 more sections

Key Result

Theorem 1.2.1

There exist only finitely many congruence Fuchsian groups up to conjugation whose associated Shimura curves are hyperelliptic. Moreover, this finite set is contained in a computable set of explicit arithmetic Fuchsian groups. Here, an explicit arithmetic Fuchsian groups is given by an explicit finit

Theorems & Definitions (147)

  • Theorem 1.2.1
  • Conjecture 1.2.2
  • Conjecture 1.3.1
  • Theorem 1.3.2
  • Proposition 2.1.1
  • proof
  • Lemma 2.1.2: KV
  • Proposition 2.1.3
  • proof
  • Lemma 2.1.4
  • ...and 137 more