The PI studies two problems which come from the study of residual automorphic representations: First, poles of completed automorphic L-functions. Second, parametrization of the local components of residual automorphic representations. It is a part of Arthur's conjecture on parametrizing the discrete spectrum in terms of representations of a certain profinite arithmetic group. The first problem is to establish the holomorphy of all completed automorphic L-functions attached to generic cuspidal representations which appear in the constant terms of Eisenstein series. The PI uses Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations. This is possible because one has the holomorphy and non-vanishing of normalized local intertwining operators due to recent progress on local results, such as standard module conjecture and Shahidi's conjecture on the holomorphy of local L-functions. For example, the PI established, as a joint work with Shahidi, the holomorphy of the third symmetric power L-functions attached to non-monomial cuspidal representations of GL_2, which had been unsolved last 20 years. The PI continues to work on the existence of the symmetric cube lift of cuspidal representations of GL_2, using the converse theorem. The second problem is to parametrize the local components of residual automorphic representations. This amounts to parametrizing the image of normalized local intertwining operators. The PI wants to show that the local components of the residual automorphic representations coming from the trivial character of the torus, are parametrized by the distinguished unipotent orbits of the dual group and Springer correspondence.
This research area is in a part of number theory generally known as the Langlands program. Number theory is the study of the properties of whole numbers and is the oldest branch of mathematics. From the beginning, problems in number theory have served as a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.
