Papers › Counting intersection numbers of closed geodesics on Shimura curves
Counting intersection numbers of closed geodesics on Shimura curves
James Rickards
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Let Γ⊆PSL(2, ℝ) correspond to the group of units of norm $1$ in an Eichler order O of an indefinite quaternion algebra over ℚ. Closed geodesics on Γ\ℍ correspond to optimal embeddings of real quadratic orders into O. The weighted intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon-Vonk on a real quadratic analogue to the difference of singular moduli. In this paper, we study the total intersection number over all embeddings of a given pair of discriminants. We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection. This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing the difference of singular moduli. The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper.
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