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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Superlacunary cusp forms

Authors: Ken Ono and Sinai Robins
Journal: Proc. Amer. Math. Soc. 123 (1995), 1021-1029
MSC: Primary 11F37; Secondary 11F11, 11F32
MathSciNet review: 1242101
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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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Additional Information

PII: S 0002-9939(1995)1242101-2
Keywords: Modular forms, lacunarity, superlacunarity
Article copyright: © Copyright 1995 American Mathematical Society

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