On Shimura's decomposition

TL;DR

This paper discusses Shimura's decomposition of half-integral weight cusp forms, which is crucial for understanding the relationship between L-function values and modular form coefficients.

Contribution

It provides an explicit computation method for Shimura's decomposition, aiding practical applications in number theory.

Findings

Explicit decomposition formulas are derived.

Enhanced computational techniques for Shimura's decomposition.

Facilitates practical applications of Waldspurger's theorem.

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 $S_{k/2}(N,\chi,\phi)$ where $\phi$ 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.