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Irregular cusps of ball quotients

Yota Maeda

Abstract

We study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma.

Irregular cusps of ball quotients

Abstract

We study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma.

Paper Structure

This paper contains 30 sections, 25 theorems, 143 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. 0-dimensional cusps
  3. Irregular cusps
  4. Case of ${\mathbb Q}(\sqrt{-1})$
  5. Case of ${\mathbb Q}(\sqrt{-3})$
  6. Other cases
  7. Relation with irregular cusps on orthogonal modular varieties
  8. Discriminant kernel case
  9. Preparation
  10. Case of ${\mathbb Q}(\sqrt{-1})$
  11. Case of ${\mathbb Q}(\sqrt{-3})$
  12. Other cases
  13. Ramification divisors and canonical singularities
  14. Low slope cusp form trick
  15. A ball quotient of non-negative Kodaira dimension
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $F$ be an imaginary quadratic field and $L$ be a Hermitian lattice of signature $(1,n)$ over $\mathscr{O}_F$. For a finite index subgroup $\Gamma\subset\mathrm{U}(L)({\mathbb Z})$, we assume that there is a non-zero cusp form $\Psi$ of weight $k$ with respect to $\Gamma$ on $D_L$. In addition, w Then the ball quotient $\mathscr{F}_L(\Gamma)$ is of general type.

Figures (7)

  • Figure 1: $F\neq{\mathbb Q}(\sqrt{-1}),{\mathbb Q}(\sqrt{-3})$
  • Figure 2: $F={\mathbb Q}(\sqrt{-1})$
  • Figure 3: $F={\mathbb Q}(\sqrt{-3})$
  • Figure 4: Orthogonal case
  • Figure 5: Relationship between unitary and orthogonal for $F\neq{\mathbb Q}(\sqrt{-1}),{\mathbb Q}(\sqrt{-3})$
  • ...and 2 more figures

Theorems & Definitions (61)

  • Theorem 1.1: Low slope cusp form trick, Theorem \ref{['low_slope_trick_unitary']}
  • Remark 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • ...and 51 more