The Manin-Drinfeld theorem and the rationality of Rademacher symbols

Abstract

For any noncocompact Fuchsian group Γ, we show that periods of the canonical differential of the third kind associated to residue divisors of cusps are expressed in terms of Rademacher symbols for Γ - generalizations of periods appearing in the classical theory of modular forms. This result provides a relation between Rademacher symbols and the famous theorem of Manin and Drinfeld. On this basis, we present a straightforward group-theoretic argument to establish both the rationality of Rademacher symbols and the validity of the Manin-Drinfeld theorem for new families of Fuchsian groups and algebraic curves.