The cuspidal group and special values of 𝐿-functions

Abstract

The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of L 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 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.