Hyperelliptic Curves and Newform Coefficients

TL;DR

This paper investigates which integers can appear as Fourier coefficients of newforms, establishing non-existence results for certain coefficients, and introduces methods linking newform values to perfect powers in recurrence sequences.

Contribution

It provides new non-existence results for Fourier coefficients of newforms, including the tau-function, and develops a novel approach connecting these questions to perfect powers in recurrence sequences.

Findings

Tau-function coefficients do not equal certain small primes or powers.

For most divisors j of 2k-1, certain coefficients are not j-th powers.

Conditional results on coefficients involving higher powers and conjectures.

Abstract

We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes $\ell < 100$ and all integers $m \geq 1$, we have $$ \tau(n) \neq \pm \ell, \pm 5^m. $$ For general newforms $f$ with even integer weight $2k$ and integer coefficients, we prove for most integers $j$ dividing $2k-1$ and all ordinary primes $p$ that $a_f(p^2)$ is never a $j$-th power. We prove a similar result for $a_f(p^4)$, conditional on the Frey-Mazur Conjecture. Our primary method involves relating questions about values of newforms to the existence of perfect powers in certain binary recurrence sequences, and makes use of bounds from the theory of linear forms in logarithms. The method extends without difficulty to a large family of Lebesgue-Nagell equations with fixed exponent. To prove results about general newforms, we also make use of the modular method and Ribet's level-lowering theorem.