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Register a new userModuli of Fontaine--Laffaille representations and a mod-$p$ local-global compatibility result
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Let $F/F^+$ be a CM field and let $widetilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $overline{r}: mathrm{Gal}(overline{mathbb{Q}}/F)rightarrow mathrm{GL}_n(overline{mathbb{F}}_p)$ be a continuous representation which we assume to be modular for a unitary group over $F^+$ which is compact at all real places. We prove, under Taylor--Wiles hypotheses, that the smooth $mathrm{GL}_n(F_{widetilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p$ cohomology with infinite level above $widetilde{v}|_{F^+}$ determines $overline{r}|_{mathrm{Gal}(overline{mathbb{Q}}_p/F_{widetilde{v}})}$, when this latter restriction is Fontaine--Laffaille and has a suitably generic semisimplification.
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Let $rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $mathrm{Gal}(overline{mathbb{Q}_p}/mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2 e 0$ on $D_{mathrm{st}}(rho_p)$. When $rho_p$ comes from an automorphic representation $pi$ of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+}^infty)$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $mathrm{GL}_3(mathbb{Q}_p)$ which only depends on and completely determines $rho_p$.
We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / QQ_p$ we will construct a bijection [ CL_g : CA^0_g(G_2,K) rightarrow CG^0(G_2,K) ] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $rho : W_K to G_2(CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.
Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$ and $overline{r} : {rm Gal}(overline F/F)rightarrow mathrm{GL}_2(overline{mathbb{F}}_p)$ an irreducible modular continuous Galois representation which, at the place $v$, is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of $mathrm{GL}_2(F_v)$ over $overline{mathbb{F}}_p$ associated to $overline{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:mathbb{Q}]$, as well as several related results.
We analyze reducibility points of representations of $p$-adic groups of classical type, induced from generic supercuspidal representations of maximal Levi subgroups, both on and off the unitary axis. We are able to give general, uniform results in terms of local functorial transfers of the generic representations of the groups we consider. The existence of the local transfers follows from global generic transfers that were established earlier.
Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $pi$ of $G$, determine the occurrence of an irreducible representation $sigma$ of $H$ in the restriction of $pi$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $sigma$ of $H$, find an irreducible representation $pi$ of $G$ such that $sigma$ occurs in the restriction of $pi$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [JZ15]. The method may be applied to other classical groups as well.
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