Ternary Quadratic Forms And Half-Integral Weight Modular Forms

TL;DR

This paper constructs explicit bases for certain half-integral weight modular forms linked to newforms with non-zero central L-values, using Waldspurger's results and local considerations.

Contribution

It provides a method to explicitly compute bases for specific subspaces of half-integral weight modular forms associated with non-zero central L-values.

Findings

Explicit bases for the subspace $S_{k/2}(\Gamma_0(4N),F)$ are constructed.

The approach leverages Waldspurger's theorem and local analysis.

The method applies to forms with $L(F,1/2) eq0$.

Abstract

Let $k$ be a positive integer such that $k\equiv3\mod4$, and let $N$ be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace $S_{\frac{k}{2}}(\Gamma_{0}(4N),F)$ of half-integral weight modular forms associated, via the Shimura correspondence, to a newform $F\in S_{k-1}(\Gamma_{0}(N))$, which satisfies $L(F,\frac{1}{2})\neq0$. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given $F$ via local considerations, once a form in the Kohnen space has been determined