# 학과 세미나 및 콜로퀴엄

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자연과학동 (E6-1), Room 3438
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 3438

정수론

The aim of the lecture series is twofold:
(1) give an overview of the “classical” theory of Breuil-Kisin modules, and
(2) discuss its application to the construction of certain p-adic Galois deformation rings.
In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition.
Here is the time and venue for each talk:
13 Feb (Mon): 3 talks
*) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실)
*) 16:00 -17:30 at E6-1, Rm 3434
14 Feb (Tue): 1 talk
*) 16:15 -17:45 at E6-1, Rm 3434
15 Feb (Wed) to 16 Feb (Thu):
*) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실)
*) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture)
17 Feb (Fri)
*) 10-11:30 at E6-1, Rm 3438
*) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

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자연과학동 (E6-1), Room 1401 (최석정강의실)
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 1401 (최석정강의실)

정수론

####
자연과학동 (E6-1), Room 1401 (최석정강의실)
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 1401 (최석정강의실)

정수론

####
자연과학동 (E6-1), Room 1401 (최석정강의실)
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 1401 (최석정강의실)

정수론

####
자연과학동 (E6-1), Room 1401 (최석정강의실)
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 1401 (최석정강의실)

정수론

####
자연과학동 (E6-1), Room 3434
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 3434

정수론

####
자연과학동 (E6-1), Room 1401 (최석정강의실)
정수론
(김완수(KAIST), 박철(UNIST))
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 1401 (최석정강의실)

정수론

####
자연과학동 (E6-1), Room 1401 (최석정강의실)
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 1401 (최석정강의실)

정수론

####
자연과학동 (E6-1), Room 3434
정수론
김완수(KAIST), 박철(UNIST)
Introduction to Breuil-Kisin modules and application to Galois deformation theory

자연과학동 (E6-1), Room 3434

정수론

Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

Affine Deligne-Lusztig varieties show up naturally in the study of Shimura varieties, Rapoport-Zink spaces, and moduli spaces of local shtukas. Among various questions on its geometric properties, the question on the connected components turns out to be a fairly important problem. For example, Kisin, in his proof of the Langlands-Rapoport conjecture (in a weak sense) for abelian type Shimura variety with the hyperspecial level structure, crucially used the description of the set of connected components. Since then, many authors have answered this question in various restricted cases. I will first discuss what is the conjectural description of the connected components and related previous works. Then, I will explain my new result (joint work with Ian Gleason and Yujie Xu) which finishes the question in the mixed characteristic case and, if time permits, new ingredients.

Affine Deligne-Lusztig varieties are first defined by Rapoport as the (conjectural) p-part of the so-called Langlands-Rapoport conjecture. It can be understood as a p-adic generalization of the classical Deligne-Lusztig varieties. One of the most basic questions is 'when they are nonempty'. For a certain union, the nonemptiness criterion is completely known (by the so-called Mazur's inequality or B(G,μ)). However, the question about the "individual" ones is moderately open (with no general conjecture). I will discuss old and new nonemptiness results and suggest a new conjecture, for the individual ones, in the basic case. As an application, I will briefly mention a new explicit dimension formula in the rank 2 case (for which no conjectural formula was stated before).

The converse theorem for automorphic forms has a long history beginning with the work of Hecke (1936) and a work of Weil (1967): relating the automorphy relations satisfied by classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. The classical converse theorems were reformulated and generalised in the setting of automorphic representations for GL(2) by Jacquet and Langlands (1970). Since then, the converse theorem has been a cornerstone of the theory of automorphic representations.
Venkatesh (2002), in his thesis, gave new proof of the classical converse theorem for modular forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this talk, we extend Venkatesh’s proof of the converse theorem to forms of arbitrary levels and characters with the gamma factors of the Selberg class type.
This is joint work with Andrew R. Booker and Michael Farmer.