The thesis of Akshay Venkatesh obtains a "Beyond Endoscopy" proof of stable functorial transfer from tori to SL2, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.
Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces (Vi,Qi) of even dimension by the equation Q1(v1)=Q2(v2)=Q3(v3). I will sketch the proof of this formula in the first portion of the talk. In the second portion, time permitting, I will discuss how these summation formulae lead to functional equations for period integrals for automorphic representations of GLn1 × GLn2 × GLn3 where the ni are arbitrary, and speculate on the relationship between these period integrals and Langlands L-functions.
I will describe applications of a 6-dimensional string theory to the Geometric Langlands Programme and
to the Knot Categorification Programme.
Let G be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of G. The parameter is a homomorphism of the global Galois group into the Langlands L-group LG of G. I will report on my joint work in progress with Böckle, Khare, and Thorne on the Taylor-Wiles-Kisin method in the setting of Lafforgue's correspondence. New (representation-theoretic and Galois-theoretic) issues arise when we seek to extend the earlier work of Böckle and Khare on the case of GLn to general reductive groups. I describe hypotheses on the Langlands parameter that allow us to apply modularity arguments unconditionally, and I will state a potential modularity theorem for a general split adjoint group.
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