Serre Weight Conjectures for $p$-adic Unitary Groups (19rit273)


(Université Paris 8)

Karol Koziol (University of Alberta)


The Banff International Research Station will host the "Serre weight conjectures for $p$-adic unitary groups" workshop in Banff from July 7, 2019 to July 14, 2019.

The Langlands program is a wide reaching series of conjectures predicting that classical objects from harmonic analysis (e.g. modular forms) can be characterized using symmetries of algebraic equations (e.g. Galois representation). Among its most spectacular materialization is the proof of Fermat's Last Theorem and the Shimura--Taniyama--Weil conjecture (the celebrated statement that "all elliptic curves over Q are modular'').

Such results require the study of congruences between modular forms and Galois representations, following an insight of J-P. Serre in the 70s. However, it was not until the work of Breuil in the early 2000s that a precise formulation of a "mod p and p-adic Langlands program" was made for the group GL2, with subsequent applications to the Fontaine-Mazur conjecture and the Breuil-Mézard conjecture.

GL2 is however too specific to fully describe symmetries that appear in the nature of automorphic forms. Recent generalizations in Serre weight conjectures led to new speculations involving forms of GLn, which are central objects in one of the deepest aspects of the Langlands program: the principle of functoriality. In our previous work we used a mix of classical functoriality for unitary groups in 2 variables and new techniques in deformations of Galois representations to obtain instances of a "mod p functoriality" principle: this led us to new modularity results and Serre's weight conjectures by transferring the current results known for GL2. During our proposed stay at BIRS we aim to formulate a systematic approach for mod p functoriality, and use the recent advances in modularity for GLn to obtain new results towards a still unexplored mod p and p-adic Langlands program for unitary groups.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).