Under two regulator nonvanishing hypotheses, the eigencurve at a p-irregular weight-one point is the four-branch ring with all cross terms zero, and the ordinary etale cohomology of the modular tower is not Hecke-free there.
A geometric view on I wasawa theory
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The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators
Under two regulator nonvanishing hypotheses, the eigencurve at a p-irregular weight-one point is the four-branch ring with all cross terms zero, and the ordinary etale cohomology of the modular tower is not Hecke-free there.