IMath Seminar: Dr. Ramla Abdellatif and Paul Rabiot

You are all invited to a seminar featuring talks from two visiting researchers: Dr. Ramla Abdellatif and Mr. Paul Rabiot. This is an in-person event and it will be held on Friday, August 18, 2023, 2:00 PM at the Math Building Conference Room.

Dr. Ramla Abdellatif (Université de Picardie Jules Verne)
Title: An introduction to \(p\)-modular Langlands correspondences
Abstract: Given a prime integer \(p\), Langlands correspondences for \(p\)-adic groups can be seen as non-abelian generalisations of class field theory, and involve representations with coefficients in various fields: either over the field of complex numbers (in which case we call them classical correspondences) or over the field of \(l\)-adic numbers for an arbitrary prime \(l\) (in which we call them \(l\)-modular correspondences).
The last instance is motivated by deep arithmetic and number-theoretic questions, but it is far from being understood, especially when \(l=p\). In this setting, even what happens for basic groups such as \(\mathrm{GL}_2(F)\) or \(\mathrm{SL}_2(F)\), \(F\) being a non-trivial finite extension of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers, remains really mysterious, for it requires to understand irreducible smooth representations of such groups over fields of characteristic \(p\).
This talk aims to make this rough picture more precise, to give a state of art of what is known so far, and to explain our contributions to the questions arising in this context. No specific knowledge of representation theory nor on \(p\)-adic groups is required, since we will mostly focus on the case of the special linear group \(\mathrm{SL}_2\) and give all necessary definitions.

Mr. Paul Rabiot (Université de Picardie Jules Verne)
Title: Distinction problems for \(p\)-modular representations of \(p\)-adic groups
Abstract: Let \(p\) be a prime integer and \(F\) be a finite extension of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. Understanding \(p\)-modular representations of \(p\)-adic groups, such as representations of \(\mathrm{SL}_2(F)\) of \(\mathrm{GL}_2(F)\) with coefficients in a (finite) field of characteristic \(p\), remains a very hard problem, for which very little is known so far. Such question is motivated, as for complex representations, by number-theoretic problems and stands at the core of a (conjectural) local \(p\)-modular Langlands correspondence for these groups.
In the complex setting, these representations are also studied under the light of what is called distinction, which means that they carry non-trivial linear forms. The existence of such forms and the dimension of the space of these forms (when non-zero) is at the core of several conjectures (such as those stated by Gan-Gross-Prasad, or Prasad-Savin) and is a very active domain of research. Nevertheless, when it comes to representations over fields of positive characteristic, the topic is very fresh and extremely little is known.
In this talk, we will explain more precisely the content of the previous paragraph, then we will present the problems we are interested in. If time allows it, we will state some preliminary results we obtained in this context in an ongoing joint work with. R. Abdellatif.