On the gaps between non-zero Fourier coefficients of eigenforms with CM

TL;DR

This paper investigates the distribution of non-zero Fourier coefficients of CM eigenforms, establishing bounds on the gaps between non-zero coefficients and demonstrating the existence of infinitely many such forms with controlled coefficient growth.

Contribution

It provides new bounds on the gaps between non-zero Fourier coefficients of CM eigenforms and constructs infinitely many forms with coefficients bounded by a power of n.

Findings

Non-zero Fourier coefficients occur within short intervals of size proportional to X^{1/4}.

Infinitely many CM eigenforms have Fourier coefficients bounded by n^{1/4}.

Results apply to eigenforms of level N > 1 and weight k > 2.

Abstract

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ of conductor $N$ with complex multiplication (CM) by $\mathbb{Q}(i)$, and $f_E$ is the corresponding cuspidal Hecke eigenform in $S^{\mathrm{new}}_2(\Gamma_0(N))$. Then $n$-th Fourier coefficient of $f_E$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$ for all $X \gg 0$ and for some $c > 0$. As a consequence, we produce infinitely many cuspidal CM eigenforms $f$ level $N>1$ and weight $k > 2$ for which $i_f(n) \ll n^{\frac{1}{4}}$ holds, for all $n \gg 0$.