On $L$-Functions of Modular Elliptic Curves and Certain $K3$ Surfaces

TL;DR

This paper investigates the values of Fourier coefficients of modular forms related to elliptic curves and K3 surfaces, providing explicit bounds and ruling out certain coefficients under GRH, advancing understanding of these special functions.

Contribution

The authors extend Lucas sequence methods to weight 2 and 3 newforms, deriving explicit bounds on Fourier coefficients and applying these to elliptic curves and K3 surfaces, with conditional results assuming GRH.

Findings

Identified specific possible Fourier coefficients for certain weight 3 newforms.

Provided bounds on coefficients under unconditionally and assuming GRH.

Ruling out many potential coefficients, narrowing the possibilities for these modular forms.

Abstract

Inspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $\tau$-function, one may ask whether an odd integer $\alpha$ can be equal to $\tau(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $k\geq 4$. We use these methods for weight $2$ and $3$ newforms and apply our results to $L$-functions of modular elliptic curves and certain $K3$ surfaces with Picard number $\ge 19$. In particular, for the complete list of weight $3$ newforms $f_\lambda(z)=\sum a_\lambda(n)q^n$ that are $\eta$-products, and for $N_\lambda$ the conductor of some elliptic curve $E_\lambda$, we show that if $|a_\lambda(n)|<100$ is odd with $n>1$ and $(n,2N_\lambda)=1$, then \begin{align*} a_\lambda(n) \in \,& \{-5,9,\pm 11,25, \pm41, \pm 43, -45,\pm47,49, \pm53,55, \pm59, \pm61, \pm 67\}\\ & \,\,\, \cup \, \{-69,\pm 71, \pm 73,75, \pm79,\pm81, \pm 83, \pm89,\pm 93 \pm 97, 99\}. \end{align*} Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving \begin{align*} a_\lambda(n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{align*}