A class number relation over function fields
Let \(G(m)= \sum_{dd'=m} \max d,d'\) and let \(H(d)\) be the Hurwitz class number for the discriminant \(d\); The classical Hurwitz relation [\textit{A. Hurwitz}, Math. Ann. 25, 157-196 (1885; JFM 17.0154.02)] states that \(G(m)= \sum_{\substack{ t\in \mathbb{Z}\\ t^2\leq 4m}} H(4m- t^2)\). This formula can be interpreted in terms of intersection multiplicities between Hecke correspondences on the surface \(X(1) \times X(1)\), where \(X(1)\) is the moduli space of generalized elliptic curves (the \(j\)-line) [see for example \textit{B. Gross} and \textit{K. Keating}, Invent. Math. 112, 225-245 (1993; Zbl 0811.11026)]. This paper develops the intersection theory of modular correspondences on Drinfeld's modular curves. As a consequence, the paper finds relations among class numbers of binary quadratic forms over \(F_q (T)\) which are analogous to the classical relations of Hurwitz. The techniques developed in this paper enable the author to prove that the singularities of the Drinfeld modular correspondence of prime level are ordinary double points; the proof follows similar lines as the classical proof once the machinery is developed. The paper consists of two main parts. In the first part, the author develops the proper setting for discussions of binary quadratic forms over \(F_q [T]\). In the second part, the author studies the intersection numbers between the Hecke correspondences on Drinfeld modular curves. The relation between class numbers and intersection numbers arises from the theory of Drinfeld modules with complex multiplication.
modular correspondences on Drinfeld's modular curves
scientific article; zbMATH DE number 809128