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The Heegner–Stark theorem and Stark–Heegner points

Elias Caeiro and Henri Darmon

Vol. 4 (2025), No. 1, 67–99
Abstract

The determination by Heegner, Baker and Stark of the complete list of imaginary quadratic orders of class number one relies critically on the theory of complex multiplication. A conjectural extension of this theory to real quadratic fields based on the notion of rigid analytic elliptic cocycles is shown to yield similar lists for some explicit families of real quadratic orders with small regulators.

Keywords
class number one problem, real quadratic fields, Stark–Heegner points, rigid analytic cocycles
Mathematical Subject Classification
Primary: 11R11, 11R42
Milestones
Received: 23 May 2024
Revised: 18 February 2025
Accepted: 19 February 2025
Published: 28 March 2025
Authors
Elias Caeiro
Ecole Normale Supérieure
Paris
France
Henri Darmon
Department of Mathematics and Statistics
McGill University
Montreal, QC
Canada