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Ramification on the eigencurve

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Robert Pollack
Boston University/MPIM
Mit, 12/06/2024 - 16:30 - 18:00
MPIM Lecture Hall
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Extra talk

Coleman's spectral halo conjecture gives a precise and simple description of the eigencurve over the outer annulus of weight space.  On the other hand, over the central region of weight space, the precise structure of the eigencurve is more mysterious.  With the exception of Emerton's thesis, which describes the lowest slope forms for the 2-adic eigencurve of tame level 1, there are virtually no examples of non-ordinary families in the literature.

In this talk, we aim to explore the eigencurve over this central region.  In particular, we will give a conjectural description of the constant-slope regions (and beyond) around p-newforms.  We will link this description both to high p-power congruences discovered by Conti and Gräf and to patterns in recent computations of p-adic Hecke eigenfields.

Joint with Kimball Martin and Anna Medvedovsky.

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