A class number relation over function fields
type: Item
instance of: scholarly article
title: A class number relation over function fields
zbMATH Open document ID: 0853.11094
author: Jiu-Kang Yu
published in: Journal of Number Theory
publication date: 1997-01-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11E41
zbMATH DE Number: 809128
zbMATH Keywords: Hurwitz class number
zbMATH Keywords: Hurwitz relation
zbMATH Keywords: intersection multiplicities
zbMATH Keywords: modular correspondences on Drinfeld's modular curves
zbMATH Keywords: binary quadratic forms
zbMATH Keywords: intersection numbers
reviewed by: Jeremy Teitelbaum
MaRDI profile type: Publication
full work available at URL: jnth.1995.1122
OpenAlex ID: W2030818724
DOI: 10.1006/JNTH.1995.1122
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description: scientific article; zbMATH DE number 809128