Trinomials, singular moduli and Riffaut's conjecture

TL;DR

This paper investigates Riffaut's conjecture that singular moduli of degree at least 3 cannot be roots of rational trinomials, proving it under GRH and providing partial unconditional results.

Contribution

The paper demonstrates that Riffaut's conjecture follows from the Generalized Riemann Hypothesis and offers partial unconditional proofs.

Findings

Riffaut's conjecture is implied by GRH.

Partial unconditional results support the conjecture.

Connections between singular moduli and polynomial roots are clarified.

Abstract

Riffaut (2019) conjectured that a singular modulus of degree $h\ge 3$ cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.