POINTS - Peking Online International Number Theory Seminar

In general, there exist $p$-adic automorphic forms of different weights with the same associated $p$-adic Galois representation. The existence of these companion forms is also predicted by Breuil's locally analytic socle conjecture in the $p$-adic local Langlands program. Under the Taylor-Wiles assumption, Breuil-Hellmann-Schraen proved the existence of all companion forms when the associated crystalline Galois representations have regular Hodge-Tate weights. In this talk, I will explain how to generalize their results to some cases when the Hodge-Tate weights are not necessarily regular. The method relies on Ding's construction of partially classical eigenvarieties and their relationships with some spaces of Galois representations.

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