Dedekind's eta-function in algebra and number theory: old and new problems
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 7-30 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we survey open problems concerning the Dedekind eta-function. These problems appear in various areas of algebra and number theory.
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G. V. Voskresenskaya. Dedekind's eta-function in algebra and number theory: old and new problems. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 7-30. http://geodesic.mathdoc.fr/item/ZNSL_2013_414_a1/

[1] K. Aierlend, M. Rouzen, Klassicheskoe vvedenie v sovremennuyu teoriyu chisel, Mir, M., 1987 | MR

[2] G. V. Voskresenskaya, “Prostranstva, soderzhaschie multiplikativnye eta-proizvedeniya”, Vestn. SamGU. Estestvennonauchn. ser., 2012, no. 6(97), 5–12

[3] G. V. Voskresenskaya, “Konechnye prostye gruppy i multiplikativnye $\eta$-proizvedeniya”, Zap. nauchn. semin. POMI, 375, 2010, 71–91 | MR | Zbl

[4] G. V. Voskresenskaya, “Konechnye gruppy i assotsiirovannye s nimi semeistva modulyarnykh form”, Matem. zametki, 87:4 (2010), 528–541 | DOI | MR | Zbl

[5] G. V. Voskresenskaya, “Giperkompleksnye chisla, sistemy kornei i modulyarnye formy”, Arifmetika i geometriya mnogoobrazii, Sb., Samara, 1992, 48–59 | MR | Zbl

[6] G. V. Voskresenskaya, “Parabolicheskie formy i konechnye podgruppy v $\mathrm{SL}(5,\mathbf C)$”, Funkts. analiz i ego pril., 29:2 (1995), 71–73 | MR | Zbl

[7] I. M. Gelfand, M. I. Graev, I. I. Pyatetskii-Shapiro, Teoriya predstavlenii i avtomorfnye funktsii, Nauka, M., 1966 | MR

[8] E. Knepp, Ellipticheskie krivye, Faktorial Press, M., 2004

[9] N. Koblits, Vvedenie v ellipticheskie krivye i modulyarnye formy, Mir, M., 1988 | MR

[10] S. Leng, Vvedenie v teoriyu modulyarnykh form, Mir, M., 1977 | MR

[11] S. Leng, Ellipticheskie funktsii, Nauka, M., 1984 | MR

[12] O. M. Fomenko, “Prilozheniya teorii modulyarnykh form k teorii chisel”, Itogi nauki i tekhniki. Algebra. Topologiya. Geometriya, 15, VINITI, M., 1977, 5–91 | MR | Zbl

[13] G. Shimura, Vvedenie v arifmeticheskuyu teoriyu avtomorfnykh funktsii, Mir, M., 1972

[14] A. G. van Asch, “Modular forms and root systems”, Math. Ann., 222 (1976), 145–170 | DOI | MR | Zbl

[15] A. J. F. Biagioli, “The construction of modular forms as products of transforms of the Dedekind eta-function”, Acta Arithm., 54:4 (1990), 273–300 | MR | Zbl

[16] K. Bringmann, K. Ono, “Identities for traces of singular moduli”, Acta Arithm., 119:4 (2005), 317–327 | DOI | MR | Zbl

[17] J. H. Bruiner, K. Ono, “The arithmetic of Borcherds exponents”, Math. Ann., 327 (2003), 293–303 | DOI | MR

[18] J. P. Buhler, Icosahedral Galois representations, Lect. Notes Math., 654, 1978 | MR | Zbl

[19] B. Cipra, “On the Shimura lift, apres Selberg”, J. Number Th., 32 (1989), 58–64 | DOI | MR | Zbl

[20] H. Cohen, J. Oesterle, “Dimensions des espaces de formes modulaires”, Lect. Notes Math., 627, 1976, 69–78 | DOI | MR

[21] J. Conway, S. Norton, “Monstrous Moonshine”, Bull. London Math. Soc., 11 (1979), 308–339 | DOI | MR | Zbl

[22] J. Conway et al., Atlas of finite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[23] B. Gross, D. Zagier, “On singular moduli”, J. reine angew. Math., 355 (1985), 191–220 | MR | Zbl

[24] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, 1997 | MR | Zbl

[25] P. Deligne, J.-P. Serre, “Formes modulaires de poids 1”, Ann. Sci. École Normale Sup., 7 (1974), 507–530 | MR | Zbl

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

[27] S. Fukuhara, “Dedekind symbols with polynomial reciprocity laws”, Math. Ann., 329 (2004), 315–324 | DOI | MR

[28] B. Gordon, D. Sinor, “Multiplicative properties of $\eta$-products”, Lect. Notes Math., 1395, 1987, 173–200 | DOI | MR

[29] B. Gordon, S. Robins, “Lacunarity of Dedekind $\eta$-products”, Glasgow Math. J., 37:1 (1995), 1–14 | DOI | MR | Zbl

[30] B. Gordon, K. Hughes, “Multiplicative properties of $\eta$-products. II”, Contemp. Math., 143 (1993), 415–430 | DOI | MR | Zbl

[31] T. Hiramatsu, “Higher reciprocity law and modular forms of weight one”, Comm. Math. Univ. St. Paul., 31 (1982), 75–85 | MR | Zbl

[32] T. Hiramatsu, Y. Mimura, “The modular equation and modular forms of weight one”, Nagoya Math. J., 100 (1985), 145–162 | MR | Zbl

[33] T. Hiramatsu, “Theory of automorphic forms of weight 1”, Adv. Stud. Pure Math., 13 (1988), 503–584 | MR | Zbl

[34] T. Hiramatsu, M. Sato, I. Takada, “On $S_3$-type modular forms of weight 1”, Math. Japonica, 32:6 (1987), 915–925 | MR | Zbl

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

[36] M. Knopp, G. Mason, “Generalized modular forms”, J. Number Th., 99 (2003), 1–18 | DOI | MR

[37] M. Koike, “On McKay's conjecture”, Nagoya Math. J., 95 (1984), 85–89 | MR | Zbl

[38] M. Koike, “Higher reciprocity law, modular forms of weight 1 and elliptic curves”, Nagoya Math. J., 98 (1985), 109–115 | MR | Zbl

[39] M. Koike, “Mathieu group $M_{24}$ and modular forms”, Nagoya Math. J., 99 (1985), 147–157 | MR | Zbl

[40] M. Koike, “Modular forms and the automorphism group of Leech lattice”, Nagoya Math. J., 112 (1988), 63–79 | MR | Zbl

[41] T. Kondo, “Examples of multiplicative $\eta$-functions”, J. Fac. Sci. Univ. Tokyo. Sect. 1A Math., 153 (1987), 133–149 | MR

[42] T. Kondo, “The automorphism group of the Leech lattice and elliptic modular functions”, J. Math. Soc. Japan, 37 (1985), 337–362 | DOI | MR | Zbl

[43] T. Kondo, T. Tasaka, “The theta functions of sublattice of the Leech lattice”, Nagoya Math. J., 101 (1986), 151–179 | MR | Zbl

[44] G. Ligozat, “Courbes modulaires de gendre 1”, Bull. Soc. Math. France, 43 (1972), 1–80 | MR

[45] G. Lusztig, Characters of reductive groups over a finite field, Ann. Math. Stud., 107, Princeton Univ. Press, 1984 | MR | Zbl

[46] N. Lygeros, O. Rozier, “A new solution to the equation $\tau(p)\equiv0\pmod p$”, J. Integer Seq., 13:7 (2010), Article ID 10.7.4 | MR | Zbl

[47] I. G. Macdonald, “Affine root systems and Dedekind's $\eta$-function”, Invent. Math., 15 (1972), 91–143 | DOI | MR | Zbl

[48] Y. Martin, “Multiplicative eta-quotients”, Trans. Amer. Math. Soc., 348 (1996), 4825–4856 | DOI | MR | Zbl

[49] Y. Martin, “On Hecke operators and products of the Dedekind $\eta$-function”, C. R. Acad. Sci. Paris, 322 (1996), 307–312 | MR | Zbl

[50] G. Mason, “Frame shapes and rational characters of finite groups”, J. Algebra, 89 (1984), 236–246 | DOI | MR

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

[52] G. Mason, “Finite groups and modular functions”, Proceedings of Symposia in Pure Math., 47 (1987), 181–207 | DOI | MR

[53] G. Mason, “Finite groups and Hecke operators”, Math. Ann., 283 (1989), 381–409 | DOI | MR | Zbl

[54] G. Mason, “On a system of elliptic modular forms attached to the large Mathieu group”, Nagoya Math. J., 118 (1990), 177–193 | MR | Zbl

[55] M. Newman, “Construction and application of a certain class of modular forms”, Proc. London Math. Soc., 7 (1956), 334–350 | MR

[56] M. Newman, “Construction and application of a certain class of modular forms. II”, Proc. London Math. Soc., 9 (1959), 373–387 | DOI | MR | Zbl

[57] W. Raji, “Generalized modular forms representable as eta products”, Acta Arithm., 129 (2007), 41–51 | DOI | MR | Zbl

[58] K. Ono, “Shimura sums related to imaginary quadratic fields”, Proc. Japan Acad. Ser. A, 70:5 (1994), 146–151 | DOI | MR | Zbl

[59] K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and $q$-series, AMS, Providence, 2004 | MR

[60] J.-P. Serre, H. Stark, “Modular forms of weight $\frac12$”, Lect. Notes Math., 627, 1977, 27–67 | DOI | MR | Zbl

[61] G. Shimura, “On modular forms of half-integral weight”, Ann. Math., 97 (1973), 440–481 | DOI | MR | Zbl

[62] J. Thompson, “Finite groups and modular functions”, Bull. London Math. Soc., 11 (1979), 347–351 | DOI | MR | Zbl

[63] G. V. Voskresenskaya, “One special class of modular forms and group representations”, J. Théor. Nombres Bordx., 11 (1999), 247–262 | DOI | MR | Zbl

[64] G. V. Voskresenskaya, “Multiplicative Dedekind $\eta$-functions and representations of finite groups”, J. Théor. Nombres Bordx., 17 (2005), 359–380 | DOI | MR | Zbl

[65] G. V. Voskresenskaya, Modular forms, Shimura sums and arithmetic of quadratic fields, MPI-preprint No 95, 2006

[66] G. V. Voskresenskaya, Finite groups associated to multiplicative $\eta$-products, MPI-preprint No 96, 2006