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 that $$\tau (n)\notin \{\pm 1,\pm 3,\pm 5,\pm 7,\pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $\tau (n)$ 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.

To appear in JNT Prime. For more general results, see arXiv:2005.10354