Programme

There will be three mini-courses by members of the Arbeitsgemeinschaft and five research talks. 

Mini-Courses

• Around the p-adic monodromy theorem
  We will give an introduction to the p-adic monodromy theorem, more specifically to the stacky formulation of Anschütz–Bosco–Le Bras–Rodríguez Camargo–Scholze.

• Locally analytic vectors in mixed characteristic
  An important notion in the representation theory of p-adic Lie groups on Q_p-vector spaces is that of local analyticity. A priori, the very definition does not make much sense for coefficient rings not containing Q_p. However, recent work by Laurent Berger and Sandra Rozensztajn ("super Hölder vectors"), and in a more general framework by Gal Porat, suggests that for many more p-adic coefficients rings, including even characteristic-p coefficient rings, there is a very promising substitute for local analyticity. We aim to present the core ideas. 

• The p-adic Jacquet–Langlands functor
  The aim is to explain the construction of Scholze's p-adic Jacquet–Langlands functors. Let p be a prime and F/Q_p a finite extension. Fix n ≥ 1 and let D/F be the central division algebra with invariant 1/n. To each admissible smooth mod p representation π of GL_n(F ), Scholze associates an étale F_p-sheaf F_π on projective space, which is equivariant for the unit group of D. Its cohomology then defines admissible smooth mod p representations for this unit group.

Research Talks

• Elmar Große-Klönne (Humboldt-Universität zu Berlin)

• Claudius Heyer (Universität Paderborn)

• Stefano Morra (Université Sorbonne Paris Nord)

• Peter Schneider (Universität Münster)

• Ziqian Yin (Bergische Universität Wuppertal)

Abstracts

• Elmar Große-Klönne: Fontaine–Laffaille theory for mod p-representations of Gal(Q_p-bar/Q_p) beyond its classical range
  Let k be a finite extension of F_p. Inside the category of étale (phi,Gamma)-modules equivalent with that of k-linear representations of Gal(Q_p-bar/Q_p), we explicitly describe a generic (asymptotically exhaustive if p grows) sub category. It is equivalent with a category whose objects are finite dimensional k-vector spaces, endowed with some additional linear structure. This latter category can be viewed as a quotient of a generic sub category of the category of Fontaine-Laffaille modules over k — with arbitrary weight intervals. In particular, we obtain a functor from that sub category to that of k-linear Gal(Q_p-bar/Q_p)-representations, thus removing the usual restriction (on the allowed weight interval) from classical Fontaine–Laffaille theory. The functor respects tensor products.

• Claudius Heyer: A 6-functor formalism for smooth mod p representations
  In this talk I will report on joint work with Lucas Mann, where we construct a full 6-functor formalism in the setting of smooth representations of p-adic Lie groups (over arbitrary discrete coefficient rings). As an application we use the formalism to construct a canonical anti-involution on derived Hecke algebras generalizing earlier work by Schneider–Sorensen.

• Stefano Morra: Looking for a mod p local Langlands correspondence
  The mod p local Langlands program aims at a parametrization of smooth mod p representations of a p-adic GLn in terms of mod p continuous n-dimensional Galois representations of the p-adic field.
  This was realized in the case of GL2(Qp) (and continuous 2-dimensional representations of Gal_{Qp}) by work of Colmez, following work of Breuil, and its compatibility with the mod p cohomology of (a tower above p of) modular curves was established by Emerton, all in the early 2000.
  The situation for GL2 over an extension of Qp, even an unramified extension, remains unclear, eventhough many partial results show that such a local correspondence should exist. In particular, one of the main questions is whether the mod p Hecke eigenspaces of a tower of Shimura curves at p are "purely local", i.e. only depend on the underlying local Galois representations and not on the global context. Indeed, such a locality would give strong evidence for the existence of a mod p local Langlands correspondence outside of GL2(Qp) compatible with global constructions.
  In this talk I will report on a joint work in progress, where we aim at constructing natural candidates for such a correspondence using tools from perfectoid geometry.
  
  This is joint work in progress with C. Breuil, F. Herzig, Y. Hu, K. Koziol, B. Schraen and S-W. Shin

• Peter Schneider: Smooth representations of amalgams
  Let G be a locally profinite group, k any field, and Mod(G) the category of smooth representations of G in k-vector spaces. Mod(G) always is a Grothendieck abelian category. But its finiteness properties depend very much on G and k. First I give a quick overview about some of the known results. Then I  explain in more detail the case where G is an amalgamation of two compact open subgroups K_1 and K_2. The basic result is the existence of a Mayer–Vietoris sequence which connects the Ext-groups of two G-representations V_1 and V_2 in the three categories Mod(G), Mod(K_1), and Mod(K_2). This implies that the category Mod(G) is locally coherent. Finally I briefly report on the case of amalgamations of more then two subgroups, which happens for a large class of p-adic groups.

• Ziqian Yin: Pro-p Iwahori–Hecke algebra and Galois moduli for SL_2
  TBA

Location

The meeting will take place at the Bergische Universität Wuppertal in Wuppertal, Germany. The lectures and talks will take place at Campus Grifflenberg, lecture hall 09 (room number G.10.02). 

Schedule

The registration will take place in front of the lecture hall. The coffee break will take place in the seminar room F.13.15.