Multiplicative Dedekind $\eta$-function and representations of finite groups

Voskresenskaya, Galina Valentinovna

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 359-380.

Dans cet article, nous étudions le problème de trouver des groupes finis tels que les formes modulaires associées aux éléments de ces groupes au moyen de certaines représentations fidèles appartiennent à des classes particulières de formes modulaires (appelées produits $\eta$ multiplicatifs). Ce problème est ouvert.

Nous trouvons des groupes métacycliques ayant cette propriété et décrivons les $p$-sous-groupes de Sylow, $p\ne 2,$ de tels groupes. Nous donnons également un aperçu des résulats reliant les produits $\eta$ multiplicatifs et les éléments d’ordre fini de $SL(5,ℂ).$

In this article we study the problem of finding such finite groups that the modular forms associated with all elements of these groups by means of a certain faithful representation belong to a special class of modular forms (so-called multiplicative $\eta -$products). This problem is open.

We find metacyclic groups with such property and describe the Sylow $p$-subgroups, $p\ne 2,$ for such groups. We also give a review of the results about the connection between multiplicative $\eta$-products and elements of finite orders in $SL(5,ℂ).$

[1] A.J.F. Biagioli, The construction of modular forms as products of transforms of the Dedekind eta-function. Acta Arith. LIV (1990), 274–300. | MR | Zbl

[2] H.S.M. Coxeter, W.O.J. Moser, Generators and relations for discrete groups. Springer-Verlag (1965), 161 pp. | MR | Zbl

[3] D. Dummit, H. Kisilevsky, J. McKay, Multiplicative products of $\eta$-functions. Contemp. Math. 45 (1985), 89–98. | Zbl

[4] B. Gordon, S. Sinor, Multiplicative properties of $\eta$-products. Lecture Notes in Math. 1395 (1989), 173–200. (Springer-Verlag) | Zbl

[5] M. Hall,jr, The theory of groups. The Macmillan Company. New York (1959). | MR | Zbl

[6] K. Harada, Another look at the Frame shapes of finite groups. J. Fac. Sci. Univ. Tokyo. Sect. IA. Math. 34 (1987), 491–512. | MR | Zbl

[7] T. Hiramatsu Theory of automorphic forms of weight 1. Advanced Studies in Pure Math. 13 (1988), 503–584. | MR | Zbl

[8] N. Ishii, Cusp forms of weight one, quartic reciprocity and elliptic curves. Nagoya Math. J. 98 (1985), 117–137. | MR | Zbl

[9] M. Koike, On McKay’s conjecture. Nagoya Math. J. 95 (1984), 85–89. | MR | Zbl

[10] T. Kondo, Examples of multiplicative $\eta$-products. Sci. Pap. Coll. Arts and Sci. Univ. Tokyo 35 (1986), 133–149 . | Zbl

[11] Y. Martin, K. Ono, Eta-quotients and elliptic curves. Proc. Amer. Math. Soc. 125 (1997), 3169–3176. | MR | Zbl

[12] G. Mason, Finite groups and Hecke operators. Math. Ann. 282 (1989), 381–409. | MR | Zbl

[13] G. Mason, $M_{24}$ and certain automorphic forms. Contemp. Math. 45 (1985), 223–244. | MR | Zbl

[14] K. Ono, Shimura sums related to imaginary quadratic fields. Proc. Japan Acad. 70 (A) (1994), 146–151. | MR | Zbl

[15] G.V. Voskresenskaya, Modular forms and representations of the dihedral group. Math. Notes. 63 (1998), 130–133. | MR | Zbl

[16] G.V. Voskresenskaya, Cusp forms and finite subgroups in $SL(5,ℂ).$ Functional Anal. Appl. 29 (1995), 71–73. | MR | Zbl

[17] G.V. Voskresenskaya, Modular forms and regular representations of groups of order 24. Math. Notes 60 (1996), 292–294. | MR | Zbl

[18] G.V. Voskresenskaya, One special class of modular forms and group representations. J. Th. Nombres Bordeaux 11(1999), 247–262. | Numdam | MR | Zbl

[19] G.V. Voskresenskaya, Metacyclic groups and modular forms. Math. Notes. 67 (2000), 18–25. | MR | Zbl

[20] G.V. Voskresenskaya, Finite groups and multiplicative $\eta$-products. Vestnik SamGU 16 (2000), 18–25. | Zbl

[21] G.V. Voskresenskaya, Abelian groups and modular forms Vestnik SamGU 28 (2003), 21–34. | MR | Zbl

[22] G.V. Voskresenskaya, Multiplicative products of Dedekind $\eta$-functions and group representations. Math. Notes. 73 (2003), 482–495. | Zbl