Variations of Lehmer's Conjecture for Ramanujan's tau-function

Abstract

We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for n>1 we prove thatτ(n)∉{±1,±3,±5,±7,±691}. This result is an example of general theorems (see Theorems 1.2 and 1.3 of [2]) for newforms with trivial mod 2 residual Galois representation. Ramanujan's well-known congruences for τ(n) allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty–Coleman method for hyperelliptic curves, and facts about certain Thue equations.