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VOL. 86 | 2020 On the rank one Gross–Stark conjecture for quadratic extensions and the Deligne–Ribet $q$-expansion principle
Samit Dasgupta, Mahesh Kakde

Editor(s) Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, Takeshi Tsuji

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.

Information

Published: 1 January 2020
First available in Project Euclid: 12 January 2021

Digital Object Identifier: 10.2969/aspm/08610243

Subjects:
Primary: 11R42
Secondary: 11F41 , 11F80 , 11R23

Keywords: $q$-expansion principle , Stark conjectures , theta series

Rights: Copyright © 2020 Mathematical Society of Japan

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