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Hecke structure of spaces of half-integral weight cusp forms

Published online by Cambridge University Press:  22 January 2016

Sharon M. Frechette
Sharon M. Frechette*
Affiliation:
Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA 02481-8203, U.S.A., [email protected]
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Abstract

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We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp forms Sk/2(4N,χ) where k and N are odd nonnegative integers with k ≥ 3, and χ is an even quadratic Dirichlet character modulo 4N. We give complete results in the case where N is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on N and χ, by giving conditions under which a newform has an equivalent cusp form in Sk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such N and χ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

Keywords

11F3711F3211F11
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 159 , 2000 , pp. 53 - 85
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

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