The Manin–Drinfeld theorem and the rationality of Rademacher symbols

Burrin, Claire

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 739-753.

Pour tout groupe fuchsien non compact $\Gamma$, on montre que les périodes de la différentielle associée aux diviseurs résiduels des pointes sont exprimées en fonction de symboles de Rademacher pour $\Gamma$ (qui généralisent les périodes de certaines formes modulaires). Ce résultat établit une relation entre les symboles de Rademacher et le célèbre théorème de Manin et Drinfeld. On verra, plus précisément, que les groupes fuchsiens dont les symboles de Rademacher sont à valeur rationnelle vérifient l’énoncé de Manin–Drinfeld. Dans un second temps, on démontre la rationalité des symboles de Rademacher pour plusieurs familles de groupes fuchsiens.

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.

Classification : 11F67, 11F20, 11F30
Mots clés : Periods of automorphic forms, Kronecker limit formula, torsion points on Jacobians, noncongruence Fuchsian groups

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