Invariant elements for 𝑝-modular representations of 𝐆𝐋₂(𝐐_{𝐩})

Morra, Stefano

doi:10.1090/S0002-9947-2013-05932-6

Invariant elements for $p$-modular representations of ${\mathbf {GL}}_{2}({\mathbf {Q}}_p)$
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Trans. Amer. Math. Soc. 365 (2013), 6625-6667 Request permission

Abstract:

Let $p$ be an odd rational prime and $F$ a $p$-adic field. We give a realization of the universal $p$-modular representations of ${\mathbf {GL}}_{2}(F)$ in terms of an explicit Iwasawa module. We specialize our constructions to the case $F={\mathbf {Q}}_p$, giving a detailed description of the invariants under principal congruence subgroups of irreducible admissible $p$-modular representations of ${\mathbf {GL}}_{2}({\mathbf {Q}}_p)$, generalizing previous work of Breuil and Paskunas. We apply these results to the local-global compatibility of Emerton, giving a generalization of the classical multiplicity one results for the Jacobians of modular curves with arbitrary level at $p$.

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