Intersections of special cycles on Shimura curves and Siegel Maass forms
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912911415312384 |
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| author | Bruinier, Jan Hendrik Li, Yingkun Möller, Martin |
| author_facet | Bruinier, Jan Hendrik Li, Yingkun Möller, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16248 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intersections of special cycles on Shimura curves and Siegel Maass forms Bruinier, Jan Hendrik Li, Yingkun Möller, Martin Number Theory Algebraic Geometry 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. |
| title | Intersections of special cycles on Shimura curves and Siegel Maass forms |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2602.16248 |