The Manin-Drinfeld theorem and the rationality of Rademacher symbols

TL;DR

This paper links Rademacher symbols to the Manin-Drinfeld theorem by showing their rationality for certain Fuchsian groups, revealing new relations between modular form periods and group properties.

Contribution

It establishes a connection between Rademacher symbols and the Manin-Drinfeld theorem, proving the rationality of Rademacher symbols for various Fuchsian groups.

Findings

Rademacher symbols relate to periods of differentials in modular forms.

Rational-valued Rademacher symbols verify the Manin-Drinfeld statement.

Rationality of Rademacher symbols established for multiple Fuchsian groups.

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.