Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA semisimple mod $p$ Langlands correspondence in families for $GL_2(mathbb{Q}_p)$
98
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules by certain $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules. Here we let $F=mathbb{Q}_p$ (and $pgeq 5$) and construct a morphism from $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters to $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuils semisimple mod $p$ Langlands correspondence for $mathbf{GL_2}(mathbb{Q}_p)$ and of Pav{s}k={u}nas parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $mathbf{GL_2}(mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-representations of the Galois group ${rm Gal}(overline{mathbb{Q}}_p/ mathbb{Q}_p)$.
rate research
Read More
Let $F$ be a non-archimedean local field with residue field $mathbb{F}_q$ and let $G = GL_2/F$. Let $mathbf{q}$ be an indeterminate and let $H^{(1)}(mathbf{q})$ be Vigneras generic pro-p Iwahori-Hecke algebra of the p-adic group $G(F)$. Let $V_{widehat{G}}$ be the Vinberg monoid of the dual group of $G$. We establish a generic version for $H^{(1)}(mathbf{q})$ of the Kazhdan-Lusztig-Ginzburg antispherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid $V_{widehat{G}}$ and establish the characteristic map in its equivariant K-theory. These generic constructions recover the classical ones after the specialization $mathbf{q} = q in mathbb{C}$. At $mathbf{q} = q = 0 inoverline{mathbb{F}}_q$, the antispherical map provides a dual parametrization of all the irreducible $H^{(1)}_{overline{mathbb{F}}_q}(0)$-modules. This work supersedes our earlier work arXiv:1907.08808. We explain the relationship between the two articles in the introduction.
Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $mathbb{F}_p$. We show that the supersingular irreducible $mathcal{H}^{(1)}$-modules of dimension 2 can be realized through the equivariant cohomology of the flag variety of the mod p Langlands dual group of $G$.
We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $rho$ on any decomposition group above $p$ is reducible with all the Jordan-Holder factors of dimension at most $2$. We also show some results on Breuils locally analytic socle conjecture in certain non-trianguline case. The main results are obtained by establishing an $R=mathbb{T}$-type result over the $mathrm{GL}_2(mathbb{Q}_p)$-ordinary families considered by Breuil-Ding.
Let $k$ be an algebraically closed field of positive characteristic $p$. We first classify the $D$-truncations mod $p$ of Shimura $F$-crystals over $k$ and then we study stratifications defined by inner isomorphism classes of these $D$-truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat $F$-decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over $k$.
We study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications to the Langlands programme: Using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet-Langlands correspondence between $operatorname{GSp}_4$ and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $operatorname{GSp}_4$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
