arXiv 2106.044732021-06math.RTmath.NT

Continuous Group Cohomology and Ext-Groups

Paulina Fust

英文摘要：

We prove that the continuous group cohomology groups of a locally profinite group $ G $ with coefficients in a smooth $ k $-representation $ π$ of $ G $ are isomorphic to the $ \mathrm{Ext}$-groups $ \mathrm{Ext}^i_G(\mathbb{1},π) $ computed in the category of smooth $ k $-representations of $ G $. We apply this to show that if $ π$ is a supersingular $ \overline{\mathbb{F}}_p $-representation of $ \mathrm{GL}_2(\mathbb{Q}_p) $, then the continuous group cohomology of $ \mathrm{SL}_2(\mathbb{Q}_p)$ with values in $ π$ vanishes. Furthermore, we prove that the continuous group cohomology groups of a $ p $-adic reductive group $ G $, with coefficients in an admissible unitary $ \mathbb{Q}_p $-Banach space representation $ Π$, are finite dimensional. We show that the continuous group cohomology of $ \mathrm{SL}_2(\mathbb{Q}_p) $ with values in non-ordinary irreducible $ \mathbb{Q}_p $-Banach space representations of $ \mathrm{GL}_2(\mathbb{Q}_p) $ vanishes.

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