In a celebrated memoir of 1860, Riemann studied a function known as the zeta function, whose properties imply an approximate formula for the number of prime numbers less than a given number. The optimal formula, still unproved, depends on the Riemann hypothesis on the location of zeros of the zeta function. Prior to the formulation of his celebrated hypothesis, Riemann stipulated a symmetry of the zeta function in the form of a functional equation. Thanks to the development of number theory following Riemann's memoir, we know that there is a large family of functions which includes the zeta function; these functions should encode numerical information about systems of polynomial equations, such as the number of their solutions modulo a prime number as the prime number varies. These functions should satisfy a generalized form of the Riemann hypothesis as well as a functional equation. In the early 1970s, Langlands proposed a visionary program predicting that all these functions can be viewed from a radically different perspective. This project will develop new tools that will aid in the development of Langlands' program.
Langlands' automorphic functoriality conjecture implies both meromorphic continuation and functional equation of L-functions. The functoriality conjecture remains largely unexplored beyond the endoscopic framework. In the early 2000s, Langlands (on one side) and Braverman-Kazhdan (on the other), formulated different general strategies to attack the functoriality conjecture beyond endoscopy and the functional equation of L-functions. This project aims to combine different elements of those proposals in a formulation of a new, possibly more realistic, strategy. The focus will be on the understanding of certain conjectural Schwartz spaces proposed by Braverman-Kazhdan, spaces of orbital integrals studied by Langlands and others, and the Fourier and Hankel transforms on those spaces. It is expected that geometry of arc spaces, which are infinite dimensional, will play a major role in these studies.