The p-adic Langlands correspondence: a constructive and algorithmic approach – CLap-CLap

The p-adic Langlands correspondence has become nowadays one of the deepest and the most stimulating research programs in number theory.
It was initiated in France in the early 2000's by Breuil and aims at understanding the relationships between the p-adic representations of p-adic absolute Galois groups on the one hand and the p-adic representations of p-adic reductive groups on the other hand. Beyond the case of GL2(Qp) which is now well established, the p-adic Langlands correspondence remains quite obscure and mysterious new phenomena enter the scene; for instance, on the $GLn(F)$-side one encounters a vast zoology of representations which seems extremely difficult to organize.

The CLap--CLap ANR project aims at accelerating the expansion of the p-adic Langlands program beyond the well-established case of
GL2(Qp). Its main originality consists in its very constructive approach mostly based on algorithmics and calculations with computers at all stages of the research process. We shall pursue three different objectives closely related to our general aim:
(1) draw a conjectural picture of the (still hypothetical) p-adic Langlands correspondence in the case of GLn,,
(2) compute many deformation spaces of Galois representations and make the bridge with deformation spaces of representations of reductive
groups,
(3) design new algorithms for computations with Hilbert and Siegel modular forms and their associated Galois representations (in which the p-adic Langlands correspondence is supposed to be realized).

This project will also be the opportunity to contribute to the development of the mathematical software SageMath and to the expansion of computational methodologies.