June 7
Title A p-adic Riemann-Hilbert functor for Zariski-constructible sheaves over rigid varieties
Speaker Yixiao Li
Abstract Let $X$ be a rigid variety over a $p$-adic number field. The Riemann-Hilbert functor for $p$-adic local systems has been constructed by Liu-Zhu. There is a modification of this construction which works for Zariski constructible sheaves, as indicated by Bhatt-Lurie in the case of algebraic varieties. In this talk, we construct a version of the Riemann-Hilbert functor, which sends Zariski-constructible sheaves on $X$ to filtered $\mathcal{D}$-modules, and show its basic properties.

Title A sheafified geometric Riemann-Hilbert correspondence
Speaker Jiahong Yu
Abstract Let $X$ be a smooth rigid variety over a local field $K$ of characteristic $0$. Due to the work by Ruochuan Liu and Xinwen Zhu, there is a geometric Riemann-Hilbert functor sending a $\mathbb{Q}_p$-local systems to vector bundles with integrable connections on a certain ringed space $\mathcal{X}$. We want to generalise this construction to an equivalence between $\mathbb{B}_{\mathrm{dR}}^+$-local systems and vector bundles with integrable connections on some certain ringed spaces. In this talk, I will prove a sheafified generalisation, say for any smooth adic space $X$ over $\mathbb{B}_{\mathrm{dR}}^+(K,K^+)$ where $(K,K^+)$ is a perfectoid Tate-Huber pair, there is a canonical isomorphism between the sheaf of isomorphic classes of vector bundles with integrable $t$-connections and the sheaf of isomorphic classes of $\mathbb{B}_{\mathrm{dR}}^+$-local systems. This is also a generalisation of Heuer's sheafified $p$-adic Simpson correspondence.

Title Period sheaves over the Fargues-Fontaine curve
Speaker Zekun Chen
Abstract Let $X$ be a smooth formal scheme over $\mathcal{O}_k$, a finite extension of $\mathbb{Q}_p$, We construct a period sheaf $\mathcal{O}\mathbb{C}^{I}$ for each closed interval $I\subseteq (0, \infty)$ that contains $1$. These period sheaves admits Poincare's lemma, and will lead to the computation of the cohomology of $B_I$. Moreover, the cohomology gives us a vector bundle on the Fargues-Fontaine curve.

June 8
Title Partial de Rham family of Hilbert modular forms
SpeakerYuanyang Jiang
Abstract We compute the Fontaine operator in the setting of Hilbert modular varieties, after taking b-cohomology, extending the work of Lue Pan in the modular curve case. As an application, we prove under some condition on weights that for overconvergent Hilbert modular forms, the partial de Rhamness condition on the Galois representation will imply the overconvergent form extends in one direction, i.e. it is “partially classical". Moreover, under generic condition, the partial classical overconvergent Hilbert modular forms vary in family.

Title Title: Locally analytic vectors in the completed cohomology of unitary Shimura curves
Speaker Tian Qiu
Abstract We use the methods introduced by Lue Pan to investigate the locally analytic vectors in the completed cohomology of Unitary Shimura curves. As some applications, we prove a classicality result on two-dimensional representation $\rho$ of $\text{Gal}_F$ such that $\rho|_{\text{Gal}_L}$ is regular $\sigma$-de Rham and it appears in the locally $\sigma$-analytic vectors of the completed cohomology, where $F$ is a number field, $L$ is a $p$-adic place of $F$ and $\sigma:L\hookrightarrow E$ is an embedding of $L$ to a sufficiently large finite extension of $\mathbb{Q}_p$. We also prove that in this case, the $\rho$-isotypic part in the locally $\sigma$-analytic vectors of the completed cohomology only depends on the local Galois representation $\rho|_{\text{Gal}_L}$. This is a joint work with Benchao Su.

Title Translations and the locally analytic $\operatorname{Ext}^1$ conjecture in the $\operatorname{GL}_2(L)$ case
Speaker Benchao Su
Abstract Let $p$ be a prime number. Let $L$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a suficiently large finite extension of $L$. Let $\rho_p$ be a $2$-dimensional $E$-linear continuous representation of $\operatorname{Gal}(\bar{L}/L)$, which is de Rham with regular Hodge-Tate weights. When $\rho_p$ is of global origin, we give a strong evidence on Breuil's locally analytic $\operatorname{Ext}^1$ conjecture for $\rho_p$. The proof is based on a detailed geometric study of the locally analytic sections on certain completed unitary Shimura curves, \`a la Lue Pan, and a geometric realization of the translation functors on locally analytic representations. If time permits, we will discuss furthur the case where the underlying Weil-Deligne representation of $\rho_p$ is irreducible.

June 9
Free discussions.