On Shimura's decomposition

Abstract

Let $k$ be an odd integer $\ge 3$ and $N$ a positive integer such that $4\mid N$. Let $\chi$ be an even Dirichlet character modulo $N$. Shimura decomposes the space of half-integral weight cusp forms $S_{k/2}(N,\chi )$ as a direct sum of $S_0(N,\chi )$ (the subspace spanned by 1-variable theta- series) and $S_{k/2}(N,\chi ,\varphi )$ where $\varphi$ runs through a certain family of integral weight newforms. The explicit computation of this decomposition is important for practical applications of a theorem of Waldspurger relating critical values of $L$-functions of quadratic twists of newforms of even weight to coefficients of modular forms of half-integral weight.

arXiv comment12 pages, to appear in the International Journal of Number Theory

Cited by in corpus (4)

• Shimura Lifts of Weakly Holomorphic Modular Forms
• Newforms of half-integral weight: the minus space counterpart
• A Note on the Fourier coefficients of half-integral weight modular forms
• Explicit application of Waldspurger's Theorem