Elliptic Stark conjectures and exceptional weight-one forms

A classical point of the Coleman–Mazur eigencurve is said to be exceptional if the map to weight space is nonétale at that point. This paper revisits the $p$ -adic elliptic Stark conjecture of Darmon et al. (Forum Math. Pi 3 (2015), art. id. e8) concerning a triple $(f, g, h)$ of classical modular forms of weights $(2, 1, 1)$ , and extends it to the setting where the $p$ -stabilised eigenform $g$ corresponds to such an exceptional point.

To Joël Bellaïche, with affection and admiration

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Keywords

Stark conjecture, triple product periods, modular forms of weight one

Mathematical Subject Classification

Primary: 11G05

Milestones

Received: 18 March 2023

Revised: 15 July 2024

Accepted: 31 July 2024

Published: 12 September 2025

Authors

Henri Darmon
Department of Mathematics and Statistics McGill University Montreal, QC Canada

Alan Lauder
Oxford University Oxford United Kingdom

Victor Rotger
Centre de Recerca Matemàtique Universitat Politecnica de Catalunya Barcelona Spain

Vol. 7, No. 3-4, 2025

Henri Darmon, Alan Lauder and Victor Rotger

Abstract

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Keywords

Mathematical Subject Classification

Milestones

Authors