In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order -recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which the structure of these modular unit groups and their associated cuspidal divisor class groups are also characterized, and a cuspidal divisor class number formula is given in terms of products of -functions and compared to the classical relative class number formula within the cyclotomic fields [6, 7].
Dans cet article, nous obtenons des invariants de classe et des groupes d’unités cyclotomiques en considérant des spécialisations d’unités modulaires. Nous construisons ces unités modulaires à partir de solutions d’équations fonctionnelles de -récurrence données par Selberg dans son travail généralisant les identités de Rogers-Ramanujan. Commme corollaire, nous donnons une nouvelle preuve d’un résultat de Zagier et Gupta, originellement considéré par Gauss, à propos des périodes de Gauss. Ces résultats proviennent pour partie de la thèse de l’auteur en 2006 [6] dans laquelle la structure de ces groupes d’unités modulaires et de leur groupe de classes de diviseurs cuspidaux associé est donnée en termes de produits de fonctions et comparée à la formule classique du nombre de classes relatives pour les corps cyclotomiques [6, 7].
@article{JTNB_2008__20_2_289_0, author = {Amanda Folsom}, title = {Class invariants and cyclotomic unit groups from special values of modular units}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {289--325}, publisher = {Universit\'e Bordeaux 1}, volume = {20}, number = {2}, year = {2008}, doi = {10.5802/jtnb.628}, zbl = {1172.11019}, mrnumber = {2477505}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.628/} }
TY - JOUR TI - Class invariants and cyclotomic unit groups from special values of modular units JO - Journal de Théorie des Nombres de Bordeaux PY - 2008 DA - 2008/// SP - 289 EP - 325 VL - 20 IS - 2 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.628/ UR - https://zbmath.org/?q=an%3A1172.11019 UR - https://www.ams.org/mathscinet-getitem?mr=2477505 UR - https://doi.org/10.5802/jtnb.628 DO - 10.5802/jtnb.628 LA - en ID - JTNB_2008__20_2_289_0 ER -
Amanda Folsom. Class invariants and cyclotomic unit groups from special values of modular units. Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 289-325. doi : 10.5802/jtnb.628. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.628/
[1] G. Andrews, An analytic proof of the Rogers-Ramanujan-Gordon identities. Amer. J. Math. 88 (1966), 844–846. | MR: 202616 | Zbl: 0147.26502
[2] G. Andrews, An introduction to Ramanujan’s ‘lost’ notebook. Amer. Math. Monthly 86 (1979), no. 2, 89–108. | Zbl: 0401.01003
[3] S. Bettner, R. Schertz, Lower powers of elliptic units. J. Théor. Nombres Bordeaux 13 (2001) no. 2, 339–351. | Numdam | MR: 1879662 | Zbl: 1003.11026
[4] W. Duke, Continued Fractions and Modular Functions. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 137–162. | MR: 2133308 | Zbl: 1109.11026
[5] H. M. Farkas, I. Kra, Theta constants, Riemann surfaces and the modular group. Graduate Studies in Mathematics 37. American Mathematical Society, Providence, RI, (2001). | MR: 1850752 | Zbl: 0982.30001
[6] A. Folsom, Modular Units. UCLA Ph.D. Thesis, 2006.
[7] A. Folsom, Modular units, divisor class groups, and the q-difference equations of Selberg. Preprint, submitted.
[8] S. Gupta, D. Zagier, On the coefficients of the minimal polynomials of Gaussian periods. Math. Comp. 60 (1993), no. 201, 385–398. | MR: 1155574 | Zbl: 0819.11062
[9] N. Ishida, N. Ishii, The equations for modular function fields of principal congruence subgroups of prime level. Manuscripta Math. 90 (1996), no. 3, 271–285. | MR: 1397657 | Zbl: 0871.11031
[10] N. Ishida, Generators and equations for modular function fields of principal congruence subgroups. Acta Arith. 85 (1998), no. 3, 197–207. | MR: 1627819 | Zbl: 0915.11025
[11] N. Ishii, Construction of generators of modular function fields. Math. Japon. 28 (1983), no. 6, 655–681. | MR: 732398 | Zbl: 0536.10016
[12] D. S. Kubert, S. Lang, Units in the modular function field. I. Math. Ann. 218 (1975), no. 1, 67–96. | MR: 437496 | Zbl: 0311.14005
[13] D. S. Kubert, S. Lang, Units in the modular function field. II. A full set of units. Math. Ann. 218 (1975), no. 2, 175–189. | MR: 437497 | Zbl: 0311.14005
[14] D. S. Kubert,S. Lang, Units in the modular function field. III. Distribution relations. Math. Ann. 218 (1975), no. 3, 273–285. | MR: 437498 | Zbl: 0311.14005
[15] D. S. Kubert, S. Lang, Units in the modular function field. Modular functions of one variable V. (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., Vol. 601, Springer, Berlin, (1977), 247–275. | MR: 453646 | Zbl: 0361.10022
[16] D. S. Kubert, S. Lang, Distributions on toroidal groups. Math. Z. 148 (1976), no. 1, 33–51. | MR: 401652 | Zbl: 0324.10021
[17] D. S. Kubert, S. Lang, Units in the modular function field. IV. The Siegel functions are generators. Math. Ann. 227 (1977), no. 3, 223–242. | MR: 498393 | Zbl: 0331.10012
[18] D. S. Kubert, S. Lang, The -primary component of the cuspidal divisor class group on the modular curve . Math. Ann. 234 (1978), no. 1, 25–44. | MR: 488819 | Zbl: 0354.12016
[19] D. S Kubert, S. Lang, Modular Units. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science] 244. Springer-Verlag, New York-Berlin, (1981). | MR: 648603 | Zbl: 0492.12002
[20] S. Lang, Elliptic Functions. Addison-Wesley Publishing Co., Reading, MA, (1973). | MR: 409362 | Zbl: 0316.14001
[21] K. Ramachandra, Some applications of Kronecker’s limit formulas. Ann. of Math. (2) 80 (1964), 104–148. | Zbl: 0142.29804
[22] L.J. Rogers, Second Memoir on the Expansion of Certain Infinite Products. Proc. London Math. Soc. 25 (1894), 318–343.
[23] L.J. Rogers, S. Ramanujan, Proof of certain identities in combinatory analysis. Cambr. Phil. Soc. Proc. 19 (1919), 211–216.
[24] R. Schertz, Construction of Ray Class Fields by Elliptic Units. J. Théor. Nombres Bordeaux 9 (1997), no. 2, 383–394. | Numdam | MR: 1617405 | Zbl: 0902.11047
[25] I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche. Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Klasse (1917), 302–321.
[26] A. Selberg, Über einge aritheoremetische Identitäten. Avh. Norske Vidensk. Akad. Oslo, I 1936, Nr. 8, 23 S.(1936); reprinted in Collected Papers, Vol. I, Springer-Verlag, Berlin, (1989). | Zbl: 0016.24701
[27] G. Shimura, Introduction to the Aritheoremetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan 11. Iwanami Shoten, Publishers, and Princeton University Press, (1971). | MR: 1291394 | Zbl: 0221.10029
[28] J.J. Sylvester, On certain ternary cubic equations. Amer. J. Math. 2 (1879), 357–381; reprinted in Collected Papers, Vol. 3, Cambridge, (1909), 325–339.
[29] L.C. Washington, Introduction to Cyclotomic Fields, 2nd Ed. Graduate Texts in Mathematics vol. 83, Springer Verlag, (1997). | MR: 1421575 | Zbl: 0966.11047
Cited by Sources: