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Mumford-type Shimura curves contained in the Torelli locus

Thomas Bouchet, Jeroen Hanselman, Andreas Pieper, Sam Schiavone

Abstract

In this article we give explicit equations for two Shimura families of genus 4 curves whose Jacobians are abelian fourfolds of Mumford type. These are the first explicit examples of abelian varieties over $\mathbb{Q}$ with endomorphism algebra $\mathbb{Z}$ and additional Hodge tensors.

Mumford-type Shimura curves contained in the Torelli locus

Abstract

In this article we give explicit equations for two Shimura families of genus 4 curves whose Jacobians are abelian fourfolds of Mumford type. These are the first explicit examples of abelian varieties over with endomorphism algebra and additional Hodge tensors.

Paper Structure

This paper contains 19 sections, 13 theorems, 65 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the article
  4. Code
  5. Acknowledgments
  6. Background
  7. Abelian varieties of Mumford type
  8. CM abelian fourfolds of Mumford type
  9. Triangle groups
  10. Discovery of the special families $C_{7,t},\, C_{9,t}$
  11. Proofs of the main theorems
  12. Construction of $\mathcal{X}_7$ and $\mathcal{X}_9$ as DM-stacks
  13. Viehweg--Zuo criterion
  14. The Shimura curves are of Mumford type
  15. Further questions
  16. ...and 4 more sections

Key Result

Theorem 2

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Figures (1)

  • Figure 1: Tesselations with $(p,q,r)=(2,3,7)$, $(2,3,9)$

Theorems & Definitions (38)

  • Conjecture 1: Coleman--Oort
  • Theorem 2: \ref{['prop:Shimura']}, \ref{['thm:main']}
  • Definition 3
  • Remark 4
  • Remark 5
  • Definition 6
  • Lemma 7
  • proof
  • Theorem 8
  • proof
  • ...and 28 more