Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. We study the smooth mod $p$ representations of $\mathrm{GL}_2(K)$ appearing in a tower of mod $p$ Hecke eigenspaces of the cohomology of Shimura curves, under mild genericity assumptions but notably no multiplicity one assumption at tame level, and prove that they are of finite length, thereby extending some recent results of Breuil, Herzig, Hu, Morra and Schraen to higher multiplicity. In a previous companion article we investigated the case where the local Galois representation attached to the Hecke eigensystem is semisimple; this article treats the non-semisimple case.
Let $L$ be a number field containing a CM field $K$. We compute an explicit current representing the $\mathrm{GL}_{\mathcal{O}_K}(\mathcal{O}_L)$-equivariant algebraic Eisenstein-Kronecker classes constructed by Kings-Sprang. In the Bianchi case, where $K$ is imaginary quadratic and $[L:K]=2$, this yields a purely alge...
Let $\mathscr{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$, and let $\mathfrak{X}$ be a proper formal scheme over $\mathscr{O}_K$ with semistable reduction. Answering a question of Fontaine and Jannsen, a classical theorem of Hyodo and Kato gives a rational identification between the log...
We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice $\Gamma$ for which $$H^1_{\m...
We show that the Tate module of the $\ell$-adic unramified cohomology $H^3_{\nr}(X, {\Q_\ell}/{\Z_\ell})$ of a smooth projective surface $X$ over a local or strictly local field $k$ of residue characteristic prime to $\ell$ vanishes under suitable conditions. We apply this to answer some questions concerning the left a...
Let $\Pi$ be a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_{3}(\mathbb{A}_{\mathbb{Q}})$ of weight $(a, 0, -a)$, and let $p$ be a prime. Generalising work of Loeffler--Williams in the nearly-ordinary case, we construct $p$-adic $L$-functions for each small slope regular $P_{1}$-refinement of $...
Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of it...
P. Bieliavsky, V. Gayral, S. Neshveyev et al.· 0 citations
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