Hecke action and the degree of the modular parameterization (Q2773322)

Let \(E\) be an elliptic curve over the rationals of conductor \(N\). Following Wiles, Taylor et al., it is known that \(E\) is modular, i.e.\ there exists a non-trivial morphism \(X_0(N)\to E\). The degree of this morphism is an ingredient in an explicit lower bound for the class number of imaginary quadratic fields. This was the starting point for a paper of \textit{D. Zagier} [Modular parametrizations of elliptic curves, Can. Math. Bull. 28, 372-384 (1985; Zbl 0579.14027)], which is being refined in the present paper. For this, the author develops a method to compute the Petterson norm of a cusp form of degree 2 on \(\Gamma_0(N)\) by looking at the values of period integrals at certain generators of \(\Gamma_0(N).\) As the degree is related to the Petterson norm of the pullback of the Néron differential of \(E\) under \(\phi\), this formula leads to a relation between the degree, the volume of \(E\), the Hecke eigenvalues of \(f\), and the period integrals under consideration.