## Mini-conference on p-adic Langlands Program

**活动时间：** 从 2023-08-16 09:00 到 17:00

**场地：** 北京国际数学研究中心，镜春园82号甲乙丙楼报告厅

**Organizers:**Yiwen Ding (BICMR) Yongquan Hu (MCM) Liang Xiao (BICMR)

**REGISTRATION ** (deadline : August 11)

**9 :00-10 :00 ** **Yitong Wang (Universite de Paris-Saclay)**

Abstract: Suppose that F is a totally real field in which p is inert. A 2-dimensional mod p represention rho of the absolute Galois group G_F cuts out a mod p represention pi of GL_2(F_p) in the cohomology of certain Shimura curves. In the work of Breuil-Herzig-Hu-Morra-Schraen, they constructed a functor D_A from certain mod p representions of GL_2(F_p) to multivariable (phi,Gamma)-modules, and a functor D_A^\otimes from mod p representations of G_{F_p} to multivariable (phi,Gamma)-modules. Then they show that D_A(pi) is isomorphic to D_A^\otimes(rho|_{F_p}) when rho|_{F_p} is semisimple and sufficiently generic. We generalize their proof to the non-semisimple case.

**10 :30- 11 :30 ****Yang Chen (Tsinghua University)**

Abstract: We determine the set of quaternionic Serre weights for generic two-dimensional mod p representation of Gal(\overline{Q}_p / K), where K is a finite unramified extension of Q_p. This is a joint work with Haoran Wang.

**13:00-14:00 ** **Yongquan Hu (MCM)**

Abstract: In the mod p Langlands program for GL_2, it is important to study the Hecke eigenspaces of mod p cohomology of Shimura curves. Inspired by the work of Breuil-Paskunas, it is conjectured that such representations have finite length and a special shape. In this talk, I will explain the proof of an upper bound of the length when the (mod p) Galois representation is reducible split and very generic. This is joint work in progress with Breuil, Herzig, Morra and Schraen.

**14:30-15:30 ** **Shanwen Wang (Renmin University)**

Abstract: We will discuss some new observation on Montreal functor for principal series. This is joint work in progress with Pierre Colmez.

**16:00-17:00 ** **Zicheng Qian (MCM)**

Astract: Let K be a finite extension of Q_p. We explicitly describe the global sections of the de Rham complex of the Drinfeld space in dimension n-1 as a representation of GL_n(K) (refining a result of Orlik), together with a quasi-isomorphism with the direct sum of its (shifted) cohomology groups. This is an ongoing joint work with Christophe Breuil.

**Date: August 17**** ****Venue: ****MCM 110**

*9:00-10:00 ***Shanxiao Huang (Peking University)**

Title: Paraboline Varieties and Associated Eigenvarieties

Abstract: We construct and study paraboline varieties, which interpolate succeesive extensions of deRham (phi,Gamma)-modules (up to twist by characters) of certain type. On the automorphic side, we construct relative eigenvarieties, and prove the existence of some local-global compatible morphisms between them.

**10:30-11:30****Zhixiang Wu (Muenster Universtiy)**

Title: TBA

Abstract: TBA

**13:00-14****:00****Benchao Su (Peking University)**

Title: Locally $\sigma$-analytic vectors in the completed cohomology of unitary Shimura curves

Abstract: Let $p$ be a prime number, and let $L$ be a finite extension of $\mathbb{Q}_p$. Let $E$ be a sufficiently large finite extension of $\mathbb{Q}_p$, and let $\Sigma$ be the set of $\mathbb{Q}_p$-embeddings of $L$ into $E$. Let $\sigma\in\Sigma$ be a fixed embedding of $L$ into $E$. We employ the methods introduced by Lue Pan to investigate the locally $\sigma$-analytic vectors in the completed cohomology of unitary Shimura curves with coefficients in $E$ attached to a unitary group with a $\mathrm{GL}_2(L)$-factor at $p$. As some applications, we prove a classicality result on regular $\sigma$-de Rham representations appear in the locally $\sigma$-analytic vectors of the completed cohomology of unitary Shimura curves. In this case, we also provide a geometric description of the locally $\sigma$-analytic representation attached to the Galois representation. This is a joint work with Tian Qiu.

*14:30-15:30***Yiwen Ding (Peking University)**

Title: Change of weights for locally analytic representations of GL2(Qp)

Abstract: Let D’ subset D be rank two (phi, Gamma)-modules, and pi(D’), pi(D) be the associated locally analytic GL2(Qp)-representations. We describe the relation between pi(D’) and pi(D).

**16**** :00-17 :00****Bin Zhao (Capital normal university)**

Title: Refined spectral halo for eigencurves

Abstract: Coleman-Mazur-Buzzard-Kilford conjecture predicted that over the boundary of the weight space, the eigencurve is a disjoint union of rigid analytic spaces which are finite flat over the weight space. This conjecture has been proved by the work of Liu-Wan-Xiao and Diao-Yao. In this talk, I will explain a joint work in progress with Yongquan Hu and Liang Xiao on a refinement of this conjecture and how it can be used to determine the p-adic slopes of all the crystabelline lifts of a reducible (local) mod p Galois representation. The new ingredient is a universal principal series type theory that interpolates classical principal series types.