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Intersections of special cycles on Shimura curves and Siegel Maass forms

Jan Hendrik Bruinier, Yingkun Li, Martin Möller

Abstract

We show that the generating series of the number of pairs of geodesics on a compact Shimura curve with given discriminants and intersection angle are coefficients of a non-holomorphic Siegel modular form, a theta lift of the constant function. This retrieves and generalizes counting results of Rickards via the Siegel-Weil formula. More generally, we study the genus two theta lift of Maass forms on this Shimura curve and prove a Fourier-Taylor expansion in terms of some generalized Whittaker functions. We also provide a geometric interpretation of all Fourier coefficients of these theta lifts in terms of averages of geodesic Taylor coefficients over special cycles.

Intersections of special cycles on Shimura curves and Siegel Maass forms

Abstract

We show that the generating series of the number of pairs of geodesics on a compact Shimura curve with given discriminants and intersection angle are coefficients of a non-holomorphic Siegel modular form, a theta lift of the constant function. This retrieves and generalizes counting results of Rickards via the Siegel-Weil formula. More generally, we study the genus two theta lift of Maass forms on this Shimura curve and prove a Fourier-Taylor expansion in terms of some generalized Whittaker functions. We also provide a geometric interpretation of all Fourier coefficients of these theta lifts in terms of averages of geodesic Taylor coefficients over special cycles.
Paper Structure (47 sections, 39 theorems, 270 equations, 2 figures)

This paper contains 47 sections, 39 theorems, 270 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Counting pairs of geodesics.
  3. A theta lift.
  4. General input and generalized Whittaker functions.
  5. Geodesic Taylor expansion.
  6. Hecke action.
  7. Related results.
  8. Acknowledgments.
  9. Geodesic cycles and Heegner points
  10. The setup
  11. Geodesic cycles
  12. Heegner points
  13. Two vectors in $L$: a case discussion
  14. (A) The positive definite case
  15. (B) The indefinite case with positive diagonal entries
  16. ...and 32 more sections

Key Result

Theorem 1.1

Consider the Shimura curve $X$ as above, and let $\tau=u+iv$ be a variable in the Siegel upper half space $\mathbb{H}_2$ of genus $2$. There is a Siegel Maass form $G(\tau)$ of genus $2$ and weight $1/2$ with the following properties. The $T$-th Fourier coefficient $G_T(v)$ for $T = \left(\right)$ i on the curve $X$. The number $|R_T|$ is a product of local factors, explicitly for $\gcd(D_1,D_2,D_

Figures (2)

  • Figure 1: Two geodesics intersecting (top left) or with a common perpendicular (top right). Geodesic through a point, perpendicular to a geodesic (bottom left) or through a second point (bottom right).
  • Figure 2: Geometry of two lattice vectors: (A) two positive vectors, intersecting (top left), (B) non-intersecting (top right), (C) a pair of a positive and a negative vector (bottom left), (D) two negative vectors (bottom right).

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • ...and 66 more