The cuspidal group and special values of 𝐿-functions

Stevens, Glenn

doi:10.1090/S0002-9947-1985-0800251-4

The cuspidal group and special values of $L$-functions
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by Glenn Stevens

Trans. Amer. Math. Soc. 291 (1985), 519-550

DOI: https://doi.org/10.1090/S0002-9947-1985-0800251-4

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Abstract:

The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of $L$-functions of weight two Eisenstein series. A new proof of a theorem of Kubert (Proposition 3.1) concerning the group of modular units is derived as a consequence of the method. The key lemma is a nonvanishing result (Theorem 2.1) for values of the “$L$-function” attached to a one-dimensional cohomology class over the relevant-congruence subgroup. Proposition 4.7 provides data regarding Eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group.

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