The Manin–Drinfeld theorem and the rationality of Rademacher symbols
Claire Burrin1
1 ETH Zürich Rämistrasse 101, 8092 Zürich, Switzerland
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 739-753.

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. 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.

Pour tout groupe fuchsien non compact Γ, on montre que les périodes de la différentielle associée aux diviseurs résiduels des pointes sont exprimées en fonction de symboles de Rademacher pour Γ (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.

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Accepted:
Published online:
DOI: 10.5802/jtnb.1225
Classification: 11F67, 11F20, 11F30
Keywords: Periods of automorphic forms, Kronecker limit formula, torsion points on Jacobians, noncongruence Fuchsian groups
Claire Burrin 1

1 ETH Zürich Rämistrasse 101, 8092 Zürich, Switzerland
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Claire Burrin. The Manin–Drinfeld theorem and the rationality of Rademacher symbols. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 739-753. doi : 10.5802/jtnb.1225. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1225/

[1] Michael Atiyah The logarithm of the Dedekind η-function, Math. Ann., Volume 278 (1987) no. 1, pp. 1-4 | Zbl

[2] Claire Burrin Generalized Dedekind sums and equidistribution mod 1, J. Number Theory, Volume 172 (2017), pp. 270-286 | DOI | Zbl

[3] Pete L. Clark; John Voight Algebraic curves uniformized by congruence subgroups of triangle groups, Trans. Am. Math. Soc., Volume 371 (2019) no. 1, pp. 33-82 | DOI | Zbl

[4] Yves Driencourt Sur les sommes de Dedekind attachées aux groupes de congruence, Bull. Soc. Math. Fr., Volume 111 (1983) no. 4, pp. 373-419 | DOI | Numdam | Zbl

[5] Vladimir G. Drinfeld Two theorems on modular curves, Funkts. Anal. Prilozh., Volume 7 (1973) no. 2, pp. 83-84

[6] William Duke; Özlem Imamoḡlu; Árpad Tóth Modular cocycles and linking numbers, Duke Math. J., Volume 166 (2017) no. 6, pp. 1179-1210 | Zbl

[7] William Duke; Özlem Imamoḡlu; Árpad Tóth Kronecker’s first limit formula, revisited, Res. Math. Sci., Volume 5 (2018) no. 2, 20, 21 pages | Zbl

[8] Étienne Ghys Knots and dynamics, Proceedings of the international congress of mathematicians (ICM) (Madrid, 2006). Volume I: Plenary lectures and ceremonies, Volume 111, European Mathematical Society, 2007, pp. 247-277 | Zbl

[9] Larry Joel Goldstein Dedekind sums for a Fuchsian group. I, Nagoya Math. J., Volume 50 (1973), pp. 21-47 | DOI | Zbl

[10] Heinz Helling Bestimung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe, Math. Z., Volume 92 (1966), pp. 269-280 | DOI | Zbl

[11] Henryk Iwaniec Spectral methods of automorphic forms, Graduate Studies in Mathematics, 53, American Mathematical Society, 2002

[12] Jay Jorgenson; Cormac O’Sullivan Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series, Nagoya Math. J., Volume 179 (2005), pp. 47-102 | DOI | Zbl

[13] Jay Jorgenson; Cormac O’Sullivan; Lejla Smajlović Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series, Res. Number Theory, Volume 6 (2020) no. 2, 22, 42 pages | Zbl

[14] Daniel S. Kubert; Serge Lang Modular units, Grundlehren der Mathematischen Wissenschaften, 244, Springer, 1981 | DOI

[15] Tomio Kubota Elementary theory of Eisenstein series, Kodansha Ltd.; John Wiley & Sons, 1973

[16] Serge Lang Introduction to algebraic and abelian functions, Graduate Texts in Mathematics, 89, Springer, 1982 | DOI

[17] Yuri I Manin Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 36 (1972), pp. 19-66 | Zbl

[18] Curt Meyer Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Mathematische Lehrbücher und Monographien. II. Abteilung: Mathematische Monographien, 5, Akademie-Verlag, 1957

[19] V. Kumar Murty; Dinakar Ramakrishnan The Manin–Drinfeld theorem and Ramanujan sums, Proc. Indian Acad. Sci., Math. Sci., Volume 97 (1987) no. 1-3, pp. 251-262 | DOI | Zbl

[20] Hans Rademacher; Emil Grosswald Dedekind sums, The Carus Mathematical Monographs, 16, The Mathematical Association of America, 1972 | DOI

[21] Michel Raynaud Courbes sur une variété abélienne et points de torsion, Invent. Math., Volume 71 (1983) no. 1, pp. 207-233 | DOI | Zbl

[22] David E. Rohrlich Points at infinity on the Fermat curves, Invent. Math., Volume 39 (1977) no. 2, pp. 95-127 | DOI | Zbl

[23] Anthony J. Scholl Fourier coefficients of Eisenstein series on non-congruence subgroups, Math. Proc. Camb. Philos. Soc., Volume 99 (1986) no. 1, pp. 11-17 | DOI | Zbl

[24] Goro Shimura Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 11, Mathematical Society of Japan, 1971

[25] Ichiro Takada Dedekind sums of Γ(N), Jpn. J. Math., Volume 12 (1986) no. 2, pp. 401-411 | DOI | Zbl

[26] Kisao Takeuchi Arithmetic triangle groups, J. Math. Soc. Japan, Volume 29 (1977) no. 1, pp. 91-106 | Zbl

[27] I. Vassileva Dedekind eta function, Kronecker limit formula and Dedekind sum for the Hecke group, Ph. D. Thesis, UMass Amherst (USA) (1996)

[28] Don Zagier Nombres de classes et fractions continues, Journées arithmétiques de Bordeaux (1974) (Astérisque), Volume 24-25, Société Mathématique de France, 1974, pp. 81-97 | Zbl

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