Superlacunary cusp forms

Ono, Ken; Robins, Sinai

doi:10.1090/S0002-9939-1995-1242101-2

Superlacunary cusp forms
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by Ken Ono and Sinai Robins

Proc. Amer. Math. Soc. 123 (1995), 1021-1029

DOI: https://doi.org/10.1090/S0002-9939-1995-1242101-2

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Abstract:

Many researchers have studied Euler product identities of weight $k = \frac {1}{2}$ and $k = \frac {3}{2}$, often related to the Jacobi Triple Product identity and the Quintuple Product identity. These identities correspond to theta series of weight $k = \frac {1}{2}$ and $k = \frac {3}{2}$, and they exhibit a behavior which is defined as superlacunary. We show there are no eigen-cusp forms of integral weight which are superlacunary. For half-integral weight forms with $k \geq \frac {5}{2}$, we give a mild condition under which there are no superlacunary eigen-cusp forms. These results suggest the nonexistence of similar Euler-Product identities that arise from eigen-cusp forms with weight $k \ne \frac {1}{2}$ or $\frac {3}{2}$.

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