The Manin-Drinfeld theorem and the rationality of Rademacher symbols

Abstract

For any noncocompact Fuchsian group $\Gamma$, 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 $\Gamma$ - 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. More precisely, Fuchsian groups whose Rademacher symbols are rational-valued verify the statement of Manin-Drinfeld. We then establish the rationality of Rademacher symbols for various families of Fuchsian groups.

Corrects false statement (Proposition 4.4) in previous version. Added references and improved exposition. Comments welcome!

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