CBGA · Cohomology of Bianchi Groups and Arithmetic

The goal of this proposal is to make a significant impact on the career of the researcher by diversifying his knowledge and skills through five training objectives and two exciting research projects on the cohomology of Bianchi groups.Bianchi groups are groups of the form SL(2,R) where R is the ring of integers of an imaginary quadratic fields. They arise naturally in the study of hyperbolic 3-manifolds and are central to the theory of Bianchi modular forms, that is, modular forms for GL(2) over imaginary quadratic fields.The research goal of this proposal is to make progress in our understanding of a central open problem in the theory of Bianchi modular forms: ``What is the arithmetic role of the torsion classes in the cohomology of congruence subgroups of Bianchi groups ?”. More generally, torsion classes in the cohomology of arithmetic groups present a phenomenon that is important to understand for Langlands Programme and the case of Bianchi groups is the simplest one that this phenomenon can be seen in. So any progress that will be made through this proposal is expected to have affect on the torsion phenomenon in general.We will attack this problem via two projects. The first project aims to reveal and study a relationship between 2-dimensional even mod p Galois representations of Q and torsion classes, while the second project aims to prove cases of a fundamental conjecture about the existence of 2-dimensional mod p Galois representations of imaginary quadratic fields associated to torsion classes.The training goal of our proposal is to equip the researcher further with advanced tools and knowledge while strengthening his existing skills. Through five training objectives, we aim to diversify the theoretical aspects of the research portfolio of the researcher by immersing him to the arithmetic theory of Siegel modular forms of genus 2, Jacquet-Langlands correspondence and its applications, p-adic modular forms and p-adic L-functions and others.""