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Multipliers of a Family of Almost Modular Functions

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Abstract

The goal of this paper is to complete an investigation begun by Cohn and Knopp in their 1994 paper, “Application of Dedekind eta-multipliers to modular equations.” The paper concerned Λ k (z), a family of modular forms on Γ0(N) (N a positive integer) with possibly non-trivial multiplier systems. Cohn and Knopp defined new functions Ψ k (z) and a new group containing Γ0(N) and proved that for all S in the larger group and for all k, Λ k (Sz) = M k(S)Ψ k (z), where M k(S)24 = 1. This yielded interesting invariance properties of Λ k , dependent on the values of M k(S). Fixing a constant integer e, independent of k, Cohn and Knopp proved that for all k and all S in the larger group, M k(S)e = (±1)e. They determined the sign of M k(S)e in many, but not all, cases. In this paper, we give a complete determination of the values of M k(S)e in the remaining cases.

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Authors and Affiliations

  1. Department of Mathematics, Bryn Mawr College, Bryn Mawr, Pennsylvania, 19010, USA

    Helen G. Grundman & Anne Schwartz

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  1. Helen G. Grundman
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  2. Anne Schwartz
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Grundman, H.G., Schwartz, A. Multipliers of a Family of Almost Modular Functions. The Ramanujan Journal 8, 47–56 (2004). https://doi.org/10.1023/B:RAMA.0000027197.23232.e2

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  • DOI: https://doi.org/10.1023/B:RAMA.0000027197.23232.e2

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