Tuesday, October 18, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Alireza Shavali (University Heidelberg)

**Abstract: **Recall the definition of the Weil group of a non-archimedean local field. Give some recollections about reductive groups, in order to define the L-group and L-homomorphisms. Give the different possible definitions [DHKM20, (1)-(4) on pages 2 and 4] of L-parameters. Explain why these agree for ℓ-adic coefficients, using Grothendieck’s ℓ-adic monodromy theorem, and why they differ for more general coefficients

Tuesday, October 25, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Sriram Chinthalagiri Venkata (University Heidelberg)

**Abstract: ** Explain how one can view L-parameters as cocycles, define the moduli functor Z1(WF0, Gˆ), and show it is representable by a scheme [DHKM20, p5]. Specializing to the tame case, construct the scheme Z1(WF0/PF , Gˆ) [DHKM20, §2.1] and study its geometry [DHKM20, §2.2].

Tuesday, November 8, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Timo Richarz (TU Darmstadt)

**Abstract: **Continue the study of the geometry of Z1(WF0/PF , Gˆ) [DHKM20, §2.3]. Explain how to define ℓ– adic continuity for more general coefficient rings, and construct the universal ℓ-adically continuous L-parameter [DHKM20, §2.4]. Time permitting, explain how instead of [DHKM20, Definition 2.11], one could deal with continuity for general coefficients using condensed mathematics as in [FS21, §VIII].

Tuesday, November 15, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Judith Ludwig (University Heidelberg)

**Abstract: **The goal of this talk is to describe moduli spaces of cocycles and their quotients by conjugation actions, 1 which will later be applied to L-parameters. More precisely, cover [DHKM20, §A.1,§A.2]. Then recall some background on GIT quotients, and explain the relation between the GIT quotient and the sheaf quotient of moduli of cocycles [DHKM20, §A.3].

Tuesday, November 22, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Jakob Burgi (University Heidelberg)

**Abstract: **Continue the study of moduli of cocycles [DHKM20, §A.4, §A.5, §A.6], and deduce [DHKM20, Theorem 3.1].

Tuesday, November 29, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Manuel Hoff (Universität Duisburg Essen)

**Abstract: **Prove [DHKM20, Theorem 3.4]. State [DHKM20, Theorem 3.12] and, time permitting, explain how one can prove it by modifying the proof of [DHKM20, Theorem 3.4], cf. [DHKM20, Remark 3.

Tuesday, December 6, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Can Yaylali (TU Darmstadt)

**Abstract: **Cover the first half of [DHKM20, §4], up to Theorem 4.13 and its proof.

Tuesday, December 13, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Patrick Bieker (TU Darmstadt)

**Abstract: **[DHKM20, §4]. The most important results are [DHKM20, Proposition 4.17, Theorem 4.18, Proposition 4.23, Theorem 4.29, Corollary 4.30].

Tuesday, December 20, 2022, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Rızacan Çiloğlu (TU Darmstadt)

**Abstract: **Cover [DHKM20, §5.1, §5.2], ending with a proof of [DHKM20, Theorem 5.5]. (In the course of the proof, it is not necessary to go through the classification of reductive groups. Instead, it is enough to illustrate what happens in general by covering the simpler cases.)

Tuesday, January 10, 2023, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Gebhard Böckle (Uinversity Heidelberg)

**Abstract: **Introduce the notion of banal primes [DHKM20, §5.3]. Describe the GIT quotient of the moduli of L-parameters by the conjugation action, first integrally [DHKM20, §6.1], and then over algebraically closed fields of banal characteristic [DHKM20, §6.2].

Tuesday, January 17, 2023, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Alireza Shavali (University Heidelberg)

**Abstract: **Explain some background on Galois deformation theory (cf. [Gee22, §2,§3], [B¨oc13, §1, §3.2]) focusing on introducing the universal framed deformation ring Rρ□ of a local Galois representation ρ : GF → GLn(Fℓ). For G = GLn, show that the completion of the local ring of the moduli space of L-parameters at an Fℓ-point x ρ recovers Rρ□ (cf. [Hel20b], [Zhu20, p.20-21]).

Tuesday, January 24, 2023, 14:00 – 15:30, Darmstadt& Zoom

** Speaker:** Torsten Wedhorn (TU Darmstadt)

**Abstract: **This talk follows [DHKM22, §2]. Cover the proof of [DHKM22, Theorem 2.3], and deduce [DHKM22, Corollaries 2.4 and 2.5].