On Hecke modules generated by eta-products and imaginary quadratic fields

Ogasawara, Takeshi

doi:10.1090/proc/13186

On Hecke modules generated by eta-products and imaginary quadratic fields
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Proc. Amer. Math. Soc. 145 (2017), 23-37 Request permission

Abstract:

We study two types of modules over the Hecke algebra generated by eta-products $\eta (\tau )\eta (N\tau )$ and $\eta (\tau )^3\eta (\lambda \tau )^3$ respectively. We give dimension formulas for them in terms of the class number of imaginary quadratic field.

References

Heng Huat Chan, Shaun Cooper, and Wen-Chin Liaw, On $\eta ^3(a\tau )\eta ^3(b\tau )$ with $a+b=8$, J. Aust. Math. Soc. 84 (2008), no. 3, 301–313. MR 2453682, DOI 10.1017/S144678870800058X
Ernst Kani, Binary theta series and modular forms with complex multiplication, Int. J. Number Theory 10 (2014), no. 4, 1025–1042. MR 3208873, DOI 10.1142/S1793042114500134
Günter Köhler, Eta products and theta series identities, Springer Monographs in Mathematics, Springer, Heidelberg, 2011. MR 2766155, DOI 10.1007/978-3-642-16152-0
Gérard Ligozat, Courbes modulaires de genre $1$, Supplément au Bull. Soc. Math. France, Tome 103, no. 3, Société Mathématique de France, Paris, 1975 (French). Bull. Soc. Math. France, Mém. 43. MR 0417060
Takeshi Ogasawara, A certain eta-quotient and the class number of an imaginary quadratic field, Int. J. Number Theory 10 (2014), no. 6, 1485–1499. MR 3248167, DOI 10.1142/S1793042114500390
Kenneth A. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977, pp. 17–51. MR 0453647
B. Schoeneberg, Bemerkungen über einige Klassen von Modulformen, Nederl. Akad. Wetensch. Proc. Ser. A 70 = Indag. Math. 29 (1967), 177–182 (German). MR 0210674