Hecke operators and subgroups of \(\Gamma(2)\) (Q1345290)

The author introduces, for each odd integer \(m\), a formal sum \(\Theta_ m\) of certain integral \(2\times 2\)-matrices with determinant \(m\), where \(\Theta_ m\) is considered to be an element in the \(\mathbb{Z}\)-algebra \(\mathbb{Z}[ \overline {M}_ 2 (\mathbb{Z})]\) over the set \(\overline {M}_ 2 (\mathbb{Z})\) of classes of integral matrices modulo multiplication by \(\pm 1\). It is shown that the \(\Theta_ m\) satisfy the usual algebraic relations between Hecke operators. The proof uses modular symbols and the theory of Hecke correspondences on modular curves for congruence groups \(\Gamma\) of even level \(N\) satisfying \(\Gamma \subset \Gamma(2)\).