Invariant elements for \(p\)-modular representations of \(\mathrm{GL}_2(\mathbb Q_p)\) (Q2855943)

The paper considers some aspects of the mod \(p\) Langlands correspondence for the group \(\mathrm{GL}_2(F)\) with \(F := \mathbb Q_p\), where \(p\) is an odd prime number. Thanks to the works of Barthel, Livné and Herzig, the \(p\)-modular representation theory of \(\mathrm{GL}_2(F)\) can somehow be reduced to the study of supersingular representations. Such a representation \(\pi\) appears as a quotient of a universal representation \(\pi(\sigma, 0)\), where \(\sigma\) is some Serre weight, that is, a smooth irreducible representation of \(\mathrm{GL}_2(\mathcal{O}_F) F^\times\).NEWLINENEWLINEIt is known that \(\pi\) is completely determined by its restrictions to \(\mathrm{GL}_2(\mathcal{O}_F)\) and to \(N\), where \(N\) is the normalizer of the Iwahori subgroup. The author obtains detailed descriptions of the restrictions of \(\pi(\sigma, 0)\) in terms of certain representations \(R_{\infty,0}(\sigma)\), \(R_{\infty,-1}(\sigma)\), etc. He is then able to deduce fine information on the invariant subspaces of \(\pi(\sigma, 0)\) under congruence subgroups. As a local-global application, a generalization of the classical multiplicity one theorem to modular curves of arbitrary level at \(p\) is obtained. This is based on a precise control of the dimensions of \(I_t\), \(K_t\)-invariants of supersingular representations, together with the \(p\)-modular Langlands correspondence of Emerton-Helm; we refer to the introduction of the paper for details.NEWLINENEWLINEThe overall style of the paper is detailed and precise, written in a passionate and reader-friendly manner.