Modular local polynomial and vanishing of L-values

Abstract

Let $f(z)$ be a weight $2k$ ($k \in \Z$) newform on $\Gamma_0(N)$. It has the Fourier expansion $f(z)=\sum_{n=1}^{\infty} a(n)q^n$ ($q=2\pi i z$). For a fundamental discriminant $D$ of an arbitrary quadratic field and $\chi_D(d) = \left(\dfrac{D}{d}\right)$ its corresponding Kronecker character, we may define the ``twisted" $L$-functions of $f$ by $L(f,D,s) := \sum_{n=1}^{\infty} \dfrac{\chi_D(n)a(n)}{n^s}.$ For $D$ coprime to $N$, the analytic continuation of $L(f,D,s)$ satisfies certain functional equations under $s\mapsto 2k-s$ and so the value $L(f,D,k)$ (which is also known as the central critical value of the function $L(f,D,s)$) or its (non-) vanishing has been the central theme of many authors, such as Kohnen-Zagier \cite{9} and Waldspurger \cite{14}.

Kohnen and Zagier \cite{9} proved that for a normalized Hecke eigenform $f$ $\in S_{2k}(\SL_2(\Z))$, there is a relation between the central critical value $L(f,D,k)$ and the Fourier coefficient of a cusp form $g=\sum_{(-1)^k n \equiv 0 \textup{ or } 1 \pmod{4}}c(n)q^n \in S_{k+\frac{1}{2}}^+(\Gamma_0(4))$ which is related to $f$ under the Shimura correspondence. In this thesis, we are going to make use of their result and a modular object called locally harmonic Maass form introduced by Bringmann, Kane and Kohnen \cite{3} to obtain an algorithm that checks whether $L(f,D,k)$ vanishes. A special case of a locally harmonic Maass form, which is called a modular local polynomial will also be studied in order to know more about such objects.

The first chapter of this thesis gives a brief history about this subject and some important results that are related to the main theorem in this study. The basic concept of how to construct the algorithm for identifying the vanishing of $L(f,D,s)$ is also included in this chapter for readers to have an idea of what is going on.

In the second chapter, there will be definitions of terms that are not explained in the introduction. There are also some preliminary lemmas which have either been proven by others or follow by a straightforward argument.

In the third chapter we will put the focus on the modular local polynomials. We will evaluate the dimension of the space (over $\C$) of such object by looking at its behaviour in fundamental domains and the coefficients of the polynomial while the functions is restricted to some connected components in $\H \setminus E_D$.

In the fourth chapter, we will explain in detail the relation of a certain cusp form $f_{k,D} \in S_{2k}$ and its corresponding locally harmonic Maass form $\mathcal{F}_{1-k,D}$. Then for any normalized Hecke eigenform $f \in S_{2k}$, we construct an operator $D_{f}$ for $f_{k,D}$ and the corresponding operator $\widetilde{D_f}$ for $\mathcal{F}_{1-k,D}$ and prove that if $\widetilde{D_f}\mathcal{F}_{1-k,D}$ is a modular local polynomial, we have $L(f,D,k)=0$ and the converse also holds.