STNB2020(34th edition)

Potentially diagonalizable representations. A conjecture.


Iván Blanco Chacón


The concept of (potential) diagonalisability was introduced by Barnet-Lamb, Gee, Geraghty and Taylor to study the automorphy of a wide class of residual representations in arbitrary dimension. More recently, Dieulefait, has envisaged an application of potential diagonalisability, to construct congruence chains between modular forms which allow to prove some instances of Langlands functoriality, for instance for tensor product representations. Of certain importance in this setting is the conjecture that the residual representation of every modular form admits infinitely many modular lifts which are crystalline and potentially diagonalisable. In the present talk we prove this conjecture after exposing several preliminaries. The proof consists in two cases: the ordinary one, which uses Hida theory, and the non-ordinary one, which needs to study certain local deformation rings which are "pasted" together following a now classical argument due to Khare and Wintenberger. This is recent joint work with Luis Dieulefait.


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