Class invariants and cyclotomic unit groups from special values of modular units

Folsom, Amanda¹ 

¹ Department of Mathematics University of California, Los Angeles Box 951555 Los Angeles, CA 90095-1555, USA

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 2, pp. 289-325.

Abstract (VO)

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 $q$-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 $L$-functions and compared to the classical relative class number formula within the cyclotomic fields [6, 7].

Résumé

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 $q$-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 $L$ et comparée à la formule classique du nombre de classes relatives pour les corps cyclotomiques [6, 7].

Détail

DOI : 10.5802/jtnb.628

Bibliographie

[1] G. Andrews, An analytic proof of the Rogers-Ramanujan-Gordon identities. Amer. J. Math. 88 (1966), 844–846. | Zbl | MR

[2] G. Andrews, An introduction to Ramanujan’s ‘lost’ notebook. Amer. Math. Monthly 86 (1979), no. 2, 89–108. | Zbl

[3] S. Bettner, R. Schertz, Lower powers of elliptic units. J. Théor. Nombres Bordeaux 13 (2001) no. 2, 339–351. | Zbl | MR | mathdoc-id

[4] W. Duke, Continued Fractions and Modular Functions. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 137–162. | Zbl | MR

[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). | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[10] N. Ishida, Generators and equations for modular function fields of principal congruence subgroups. Acta Arith. 85 (1998), no. 3, 197–207. | Zbl | MR

[11] N. Ishii, Construction of generators of modular function fields. Math. Japon. 28 (1983), no. 6, 655–681. | Zbl | MR

[12] D. S. Kubert, S. Lang, Units in the modular function field. I. Math. Ann. 218 (1975), no. 1, 67–96. | Zbl | MR

[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. | Zbl | MR

[14] D. S. Kubert,S. Lang, Units in the modular function field. III. Distribution relations. Math. Ann. 218 (1975), no. 3, 273–285. | Zbl | MR

[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. | Zbl | MR

[16] D. S. Kubert, S. Lang, Distributions on toroidal groups. Math. Z. 148 (1976), no. 1, 33–51. | Zbl | MR

[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. | Zbl | MR

[18] D. S. Kubert, S. Lang, The $p$-primary component of the cuspidal divisor class group on the modular curve $X\left(p\right)$. Math. Ann. 234 (1978), no. 1, 25–44. | Zbl | MR

[19] D. S Kubert, S. Lang, Modular Units. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science] 244. Springer-Verlag, New York-Berlin, (1981). | Zbl | MR

[20] S. Lang, Elliptic Functions. Addison-Wesley Publishing Co., Reading, MA, (1973). | Zbl | MR

[21] K. Ramachandra, Some applications of Kronecker’s limit formulas. Ann. of Math. (2) 80 (1964), 104–148. | Zbl

[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. | Zbl | MR | mathdoc-id

[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

[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). | Zbl | MR

[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). | Zbl | MR