Intersections over modular curves
Let \(\Gamma \subset S L_ 2 (\mathbb{Z})\) be a subgroup of finite index containing \(-I\), \(Y\) the quotient of the complex upper half plane by \(\Gamma\), \(X\) the associated compact Riemann surface, and \(C = X - Y\). \textit{Manin} gave a description of the relative homology group \(H_ 1 (X,C; \mathbb{Z})\) in terms of modular symbols. Here, the author describes \(H_ 1 (Y; \mathbb{Z})\) as a subquotient of the group of \(\mathbb{Z}\)-valued functions on the coset space \(\Gamma \backslash SL_ 2 (\mathbb{Z})\). He then gives a formula for the intersection of a modular symbol and a suitable function, thus determining the intersection pairing \(H_ 1 (X,C; \mathbb{Z}) \times H_ 1 (Y; \mathbb{Z}) \to \mathbb{Z}\).
scientific article; zbMATH DE number 492787