The Modular Degree and the Congruence Number of a Weight 2 Cusp Form
نویسندگان
چکیده
Let f be a weight 2 normalized newform on the congruence subgroup Γ0(N) with integral Fourier coefficients. There are two important numerical invariants attached to f : its congruence number and its modular degree. By definition, the congruence number of f is the largest integer Df such that there exists a weight 2 cusp form on Γ0(N), with integral coefficients, which is orthogonal to f with respect to the Petersson inner product and is congruent to f modulo Df . The modular degree of f is the degree deg φf of the minimal parametrization φf : X0(N) → E of the strong Weil elliptic curve E/Q associated to f via the Shimura construction. Here X0(N)/Q denotes the modular curve defined by Γ0(N). It turns out that the two quantities Df and deg φf are closely related. On the one hand we have:
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