A note on lower bounds of heights of non-zero Fourier coefficients of Hilbert cusp forms

Abstract

It is a conjecture of Atkin and Serre that for any $\epsilon > 0$, there exists a constant $c(\epsilon ) > 0$ such that the Ramanujan $\tau$-function satisfies

$$\begin{aligned} |\tau (p)| \ge c(\epsilon ) ~p^{9/2 - \epsilon } \end{aligned}$$

for all sufficiently large primes. This in particular would settle Lehmer’s folklore conjecture for almost all primes and hence is widely open. In an interesting and elegant work, R. Murty, K. Murty, and T. Shorey showed that if $\tau (n)$ is odd, then $|\tau (n)| > (\log n)^{\delta }$ for some effective absolute constant $\delta > 0$. In this short note, building upon their work, we derive lower bounds for heights of certain Fourier coefficients of primitive Hilbert cusp forms. The extension to Hilbert modular forms brings in some new difficulties which are absent for the Ramanujan $\tau$-function and compel us to abandon absolute values and work with Weil heights. This brings in the further caveat that the bounds on prime powers no longer lead to bounds on arbitrary Fourier coefficients as the height inequalities are no longer compatible to derive such lower bounds. A major point in this circle of questions is the issue of non-vanishing of the eigenvalues $c({\mathfrak {p}})$ which are associated to the prime ideals ${\mathfrak {p}}$ of the ambient number field. In the final section, we indicate a somewhat curious link between this and the oscillation of the real sequence $\{c(\mathfrak {p^n})\}_{n\in {{\mathbb {N}}}}$.