Swinnerton-Dyer Type Congruences for certain Eisenstein series
نویسندگان
چکیده
This follows from the Von Staudt-Claussen Theorem regarding the divisibility of the denominators of Bernoulli numbers. Here we generalize Swinnerton-Dyer’s result to certain Eisenstein series in spaces of modular forms of weight k on a congruence subgroup of type Γ0(N) with Nebentypus character χ. These spaces are denoted by Mk(Γ0(N), χ). For background on integer weight modular forms, see [K]. Following the methods in Sections 1-3 of Chapter VII of Schoeneberg’s book, Elliptic Modular Functions , we develop the Fourier expansion at infinity of a normalized Eisenstein series EN,k,χ(z) ∈ Mk(Γ0(N), χ) which vanishes at all cusps of Γ0(N) inequivalent to
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