On the rank one Gross–Stark conjecture for quadratic extensions and the Deligne–Ribet $q$-expansion principle

Abstract

In this note, we provide a new proof of the rank 1 Gross–Stark conjecture for a quadratic extension under the assumption that there is only one prime in the base field above a rational prime $p$. The full Gross–Stark conjecture was proven by the authors in joint work with Ventullo, building on a prior result in the rank 1 setting by the first named author in joint work with Darmon and Pollack. The proof given in this note is much simpler as it does not use the theory of $p$-adic Galois cohomology and Galois representations associated to $p$-adic modular forms. Instead, the proof relies on a certain explicit construction using Theta series, congruences with Eisenstein series and the $q$-expansion principle of Deligne–Ribet.