Jacquet-Rallis transfer for parahoric functions over ramified quadratic extensions

The Gan-Gross-Prasad conjecture relates special values of certain L-functions to period integrals on classical groups. Jacquet and Rallis proposed an approach through the relative trace formula, reducing the problem to two conjectures: a fundamental lemma conjecture and a smooth transfer conjecture. In the local setting, the former is largely proven by the work of Yun and Gordon, and the second was proven by Zhang. We are interested in an arithmetic version of the problem, which aim to generalize Gross-Zagier formula, relating Neron-Tate heights of Heegner points on modular curves, to special values of derivatives of certain L-functions. Zhang proposed a relative trace formula approach, which now relies on two conjectures: an arithmetic fundamental lemma (AFL) conjecture and an arithmetic transfer (AT) conjecture. Zhang has proven AFL for low-rank unitary groups, and an example of AT was worked by Smithling-Rapoport-Zhang, although the authors could not write an explicit transfer. In this talk, I will be presenting the main objects related to this conjecture, which boils down to harmonic analysis on ramified unitary group, and I will be introducing a new conjecture aiming to realize the arithmetic transfer explicitly. This is joint work with Wei Zhang.   

Thomas Rud obtained his PhD in 2022 under the supervision of Julia Gordon. His thesis was focused on local-global principles, Galois cohomology and integrals over conjugacy orbits of p-adic groups. Since then he's been a postdoc at MIT, working with Wei Zhang on the Arithmetic Transfer Conjecture.