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Hilbert surfaces, modular forms, and Siegel-Veech constants

Duc-Manh Nguyen

Abstract

We give the values of the Siegel-Veech constants associated with saddle connections having distinct endpoints on translation surfaces in Prym eigenform loci in $Ω\mathcal{M}_3(2,2)^{\rm odd}$. In particular, we show that these constants are actually the same for all of these loci. As a by-product, we show that the Euler characteristic of the Hilbert modular surfaces which parametrize Abelian surfaces with $(1,2)$-polarization admitting a real multiplication and the Euler characteristic of their product locus are related by a simple formula. For principally polarized Abelian surfaces, a similar phenomenon has been observed by Bainbridge.

Hilbert surfaces, modular forms, and Siegel-Veech constants

Abstract

We give the values of the Siegel-Veech constants associated with saddle connections having distinct endpoints on translation surfaces in Prym eigenform loci in . In particular, we show that these constants are actually the same for all of these loci. As a by-product, we show that the Euler characteristic of the Hilbert modular surfaces which parametrize Abelian surfaces with -polarization admitting a real multiplication and the Euler characteristic of their product locus are related by a simple formula. For principally polarized Abelian surfaces, a similar phenomenon has been observed by Bainbridge.
Paper Structure (25 sections, 22 theorems, 130 equations)

This paper contains 25 sections, 22 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Statements of the results
  3. Ideas of the proof
  4. Consequences of relations of modular forms
  5. Basic definitions and properties of some modular functions
  6. Relations of modular forms
  7. Proof of Theorem \ref{['th:rel:sum:sigma:1']}
  8. Real multiplication and Hilbert modular surfaces
  9. Hilbert modular surfaces
  10. Product locus in $X_D$
  11. Prym eigenforms and Abelian surfaces with $(1,2)$-polarization
  12. Eigenforms on products of elliptic curves
  13. Euler characteristics of the curves $Q'_D$
  14. Consequences of Bainbridge's formula
  15. Proof of Theorem \ref{['th:eul:char:PD:p:XD:p']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

For all $D\in \mathbb{N}, \; D >9, D \equiv 0,1,4 \, [8]$, and $D$ is not a square, we have

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • ...and 30 more