The eigencurve at crystalline points with scalar Frobenius and Gross–Stark regulators

Abstract We give a complete description of the local geometry of the 𝑝-adic eigencurve at 𝑝-irregular classical weight one cusp forms under the assumption that certain Gross–Stark regulators do not vanish, as predicted by classical conjectures in 𝑝-adic transcendental number theory. We also provide several applications to the Hecke structure of the ordinary 𝑝-adic étale cohomology of towers of modular curves, as well as to Beilinson–Flach elements.