Bound on the Gelfand-Kirillov dimension for GL_2

Abstract: The Gelfand-Kirillov dimension of p-adic representations of a p-adic Lie group G is a non-commutative generalization of the Krull dimension for modules, to modules over Iwasawa and distribution algebras of G. As observed by Gee-Newton, knowing this dimension in the setting of completed/mod p cohomology or patching can lead to progress towards big R=T theorems. In my thesis I proved that any admissible p-adic Banach representation of GL_2K (K a p-adic field) with an infinitesimal character has GK-dimension at most [K:Q_p]. I have also proved a version in families, and a version for quaternion units. The family version can be applied to many Hecke eigenspaces of completed cohomology or to patched modules, by Dospinescu-Paškūnas-Schraen, Paškūnas-Quast, and most recently Ivančić-McDonald, who have shown the presence of infinitesimal characters in families in several cases. For example, the results of Breuil-Herzig-Hu-Morra-Schraen on GK-dimensions of smooth mod p representations of GL_2K can be generalized beyond K unramified. In my talk I intend to focus on the general result and sketch its proof in the case of GL_2Q_p. I will touch only briefly on applications, many of which still need to be worked out.

number theory

Audience: researchers in the topic

London number theory seminar

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