Relative trace formulas are a generalization of the celebrated Arthur-Selberg trace formula that are well-suited to deal with periods of automorphic forms. As with the latter, comparing different RTFs is a powerful method to establish results in the relative Langlands program. This minicourse aims to give an introduction to this wide subject. As an illustration I will present Jacquet's proof of Waldspurger results on toric periods for $\operatorname{GL}(2)$. Time permitting, I will also present the RTF approach to the Gan-Gross-Prasad conjecture for unitary groups.
The Langlands philosophy postulates a correspondence between certain automorphic representations of $\operatorname{GL}(n)$, and (compatible systems of) certain $n$-dimensional Galois representations. Congruences of automorphic forms can then be modeled by $p$-adic families of Galois representations, and one can ask how Galois theoretic constraints describe spaces of automorphic representations that reduce to the same mod $p$ Galois representation. This elementary minicourse gives an introduction to the deformation theory of Galois representations in the sense of Mazur and its relevance to modular Galois representations. The topics are:
The Poisson summation conjecture predicts that suitable spaces admit Schwartz spaces, Fourier transforms and a Poisson summation formula generalizing the familiar Poisson summation formula on a vector space. It implies the expected analytic properties of Langlands $L$-functions and thereby much of Langlands functoriality.
The lectures will focus on the following work, which are aimed at reducing a key case of the conjecture to local assertions. Several methods that can be used to construct new Poisson summation formulae from old ones will be isolated and explained.
Another relevant work in preparation is the following:
Integral representations of automorphic $L$-functions and the theory of Euler systems both provide landscapes in which to explore the many conjectures surrounding the arithmetic of special values. While these landscapes may appear to have very different origins, they share some common features. An aim of these lectures is to explain how these commonalities can be explained by multiplicity one results in local representation theory (and related properties), especially by exploring examples. We also hope to raise a few related questions.
Some level of comfort with modular forms, algebraic number theory, and Galois cohomology. Basic familiarity with automorphic forms and representations will certainly help with later lectures. As much as possible we will stick with accessible examples.
The Langlands program is a web of far-reaching and influential conjectures about connections between number theory, representation theory and geometry. Within this program, the relative Langlands program has emerged as one of its most important and productive branches. In these talks, I will give an overview of key problems in the relative Langlands program, with a focus on the elegant theory of relative Langlands duality, recently developed by Ben-Zvi, Sakellaridis, and Venkatesh.
