## One-Day Workshop on Arithmetic Geometry

**Time: **December 28, 2023

**Venue: ** Lecture Hall, Jiayibing Building, Jingchunyuan 82, BICMR

9:00-10:00

Speaker: Jiang Yuanyang (Paris-Saclay)

Title: Theta operator equals Fontaine Operator on Modular Curve

Abstract: Inspired by the work of Lue Pan, we give a new proof that for an overconvergent modular eigenform of regular weight, assuming that its associated Galois representation is irreducible, then it is classical if and only if the associated Galois representation is de Rham at p. For the proof,

we prove that theta operator coincides with Fontaine operator in a suitable sense. If time permits, we will discuss possible generalization in the Hilbert case.

10:30-11:30

Speaker: Qiu Tian (PKU)

Title: Locally analytic vectors in the completed cohomology of unitary Shimura curves

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 regular σ-de Rham representations of Gal_L appearing in the locally σ-analytic vectors of the completed cohomology, where L is a finite extension of Q_p and σ:L→E is an embedding of L to a sufficiently large finite extension of Q_p. We also prove that if a two-dimensional representation of Gal_L appears in the locally σ-algebraic vectors of the completed cohomology then it is σ-de Rham. This is a joint work with Benchao Su.

2:00-3:00

Speaker: Zhang Mingjia (Princeton)

Title: Igusa stacks and p-adic Shimura varieties

Abstract: It is conjectured by Scholze that a p-adic Shimura variety, viewed as a diamond, can be expressed as the fiber product of an adic flag variety and a so-called Igusa stack. This geometric structure has applications to understanding the cohomology of Shimura varieties and integral models. I will explain these ideas and report on progress on this conjecture.

3:30-4:30

Speaker: Tang Longke (Princeton)

Title: P1-motivic Gysin map

Abstract: Recently, Annala, Hoyois, and Iwasa have defined and studied the P1-motivic homotopy theory, a generalization of A1-motivic homotopy theory that does not require A1 to be contractible, but only requires pointed P1 to be invertible. This makes it applicable to non-A1-invariant cohomology theories such as Hodge, de Rham, and prismatic. I will recall basic facts in their theory, and construct the P1-motivic Gysin map, thus giving a uniform construction for the Gysin maps of the above cohomology theories that are automatically functorial. If time permits, I will also use this Gysin map to prove prismatic Poincaré duality.