Theta series on the Hecke groups \(G(\sqrt{2})\) and \(G(\sqrt{3})\)
For \(q=2,3\), the Hecke group \(G(\sqrt{q})\) is the group which is generated by the transformations \(z\mapsto z+\sqrt{q}\) and \(z\mapsto - 1/z\) of the upper half plane. The object of this paper are those modular forms of integral weight and an arbitrary multiplier system on \(G(\sqrt{q})\) which can be represented as a theta series with a grössencharacter (Hecke character) attached to some imaginary quadratic number field. Modular forms of this kind have been called of CM-type [\textit{K. A. Ribet}, Lect. Notes Math. 601, 17--52 (1977; Zbl 0363.10015)]. One of the reasons why they are interesting is that they can be recognized to be Hecke eigenforms without using Hecke operators: From elementary arithmetic in imaginary quadratic number fields it follows that the associated Dirichlet series have Euler product expansions. There are several infinite sequences of modular forms which are theta series with characters attached to \({\mathbb Q}(\sqrt{-3})\) for \(G(\sqrt{3})\), and to \({\mathbb Q}(\sqrt{-1})\), \({\mathbb Q}(\sqrt{-2})\) for \(G(\sqrt{2}).\) The paper contains many identities which express these theta series, for small weights, as combinations of Eisenstein series and the Dedekind eta-function \(\eta\). The theta series representations yield congruences and estimates for the Fourier coefficients of the modular forms. The estimates are slightly sharper than those which hold generally for cusp eigenforms (Ramanujan-Petersson-Deligne). Moreover, properties of the characters imply that the Fourier coefficients of otherwise unrelated modular forms coincide at certain infinite sets of primes. Results of this kind (congruences, estimates, coincidence of Fourier coefficients) are also given for the forms \(\eta^ 2\), \(\eta^ 4\), \(\eta^ 6\), \(\eta^ 8\) on the modular group. Theta series with grössencharacters on the modular group have been investigated by \textit{B. Schoeneberg} [Math. Ann. 126, 177--184 (1953; Zbl 0053.05403)], and they have been reconsidered recently by \textit{J.-P. Serre} [Glasg. Math. J. 27, 203--221 (1985; Zbl 0583.10015)].
scientific article; zbMATH DE number 4027572
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