For any noncocompact Fuchsian group , 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 (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 , on montre que les périodes de la différentielle associée aux diviseurs résiduels des pointes sont exprimées en fonction de symboles de Rademacher pour (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.
Revised:
Accepted:
Published online:
Keywords: Periods of automorphic forms, Kronecker limit formula, torsion points on Jacobians, noncongruence Fuchsian groups

@article{JTNB_2022__34_3_739_0, author = {Claire Burrin}, title = {The {Manin{\textendash}Drinfeld} theorem and the rationality of {Rademacher} symbols}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {739--753}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {34}, number = {3}, year = {2022}, doi = {10.5802/jtnb.1225}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1225/} }
TY - JOUR AU - Claire Burrin TI - The Manin–Drinfeld theorem and the rationality of Rademacher symbols JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 739 EP - 753 VL - 34 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1225/ DO - 10.5802/jtnb.1225 LA - en ID - JTNB_2022__34_3_739_0 ER -
%0 Journal Article %A Claire Burrin %T The Manin–Drinfeld theorem and the rationality of Rademacher symbols %J Journal de théorie des nombres de Bordeaux %D 2022 %P 739-753 %V 34 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1225/ %R 10.5802/jtnb.1225 %G en %F JTNB_2022__34_3_739_0
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] The logarithm of the Dedekind -function, Math. Ann., Volume 278 (1987) no. 1, pp. 1-4 | Zbl
[2] Generalized Dedekind sums and equidistribution mod 1, J. Number Theory, Volume 172 (2017), pp. 270-286 | DOI | Zbl
[3] Algebraic curves uniformized by congruence subgroups of triangle groups, Trans. Am. Math. Soc., Volume 371 (2019) no. 1, pp. 33-82 | DOI | Zbl
[4] 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] Two theorems on modular curves, Funkts. Anal. Prilozh., Volume 7 (1973) no. 2, pp. 83-84
[6] Modular cocycles and linking numbers, Duke Math. J., Volume 166 (2017) no. 6, pp. 1179-1210 | Zbl
[7] Kronecker’s first limit formula, revisited, Res. Math. Sci., Volume 5 (2018) no. 2, 20, 21 pages | Zbl
[8] 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] Dedekind sums for a Fuchsian group. I, Nagoya Math. J., Volume 50 (1973), pp. 21-47 | DOI | Zbl
[10] Bestimung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe, Math. Z., Volume 92 (1966), pp. 269-280 | DOI | Zbl
[11] Spectral methods of automorphic forms, Graduate Studies in Mathematics, 53, American Mathematical Society, 2002
[12] 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] 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] Modular units, Grundlehren der Mathematischen Wissenschaften, 244, Springer, 1981 | DOI
[15] Elementary theory of Eisenstein series, Kodansha Ltd.; John Wiley & Sons, 1973
[16] Introduction to algebraic and abelian functions, Graduate Texts in Mathematics, 89, Springer, 1982 | DOI
[17] Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 36 (1972), pp. 19-66 | Zbl
[18] Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Mathematische Lehrbücher und Monographien. II. Abteilung: Mathematische Monographien, 5, Akademie-Verlag, 1957
[19] 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] Dedekind sums, The Carus Mathematical Monographs, 16, The Mathematical Association of America, 1972 | DOI
[21] Courbes sur une variété abélienne et points de torsion, Invent. Math., Volume 71 (1983) no. 1, pp. 207-233 | DOI | Zbl
[22] Points at infinity on the Fermat curves, Invent. Math., Volume 39 (1977) no. 2, pp. 95-127 | DOI | Zbl
[23] 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] Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 11, Mathematical Society of Japan, 1971
[25] Dedekind sums of , Jpn. J. Math., Volume 12 (1986) no. 2, pp. 401-411 | DOI | Zbl
[26] Arithmetic triangle groups, J. Math. Soc. Japan, Volume 29 (1977) no. 1, pp. 91-106 | Zbl
[27] Dedekind eta function, Kronecker limit formula and Dedekind sum for the Hecke group, Ph. D. Thesis, UMass Amherst (USA) (1996)
[28] Nombres de classes et fractions continues, Journées arithmétiques de Bordeaux (1974) (Astérisque), Volume 24-25, Société Mathématique de France, 1974, pp. 81-97 | Zbl
Cited by Sources: