## 2024 Clay Research Award

### James Newton and Jack Thorne

A Clay Research Award is made to **James Newton** (Oxford) and **Jack Thorne** (Cambridge) in recognition of their remarkable proof of the existence of the symmetric power functorial lift for Hilbert modular forms.

The conjecture that the symmetric powers of automorphic representations associated to classical and Hilbert modular forms should themselves be automorphic is one of the fundamental conjectures of the programme introduced by Langlands in the late 1960’s, indeed it was cited by Langlands as a prototype test case for his conjectures. Building on earlier work of Clozel and Thorne, Newton and Thorne have written a series of ingenious papers proving this result using a very detailed application of modularity lifting results to the associated Galois representations. The proof marks a milestone in work on the Langlands programme.

### Paul Nelson

A Clay Research Award is made to **Paul Nelson** (Aarhus) in recognition of his groundbreaking contributions to the analytic theory of automorphic forms. His work has resulted in the first convexity-breaking bounds for a large class of L-functions on the critical line (including all the standard ones of GL(n)). This marks a signficant advance in a field initiated one hundred years ago by Hermann Weyl in the context of the Riemann Zeta function. Nelson analyzes L-values via certain associated automorphic periods. His powerful approach involves many ingredients, including a refinement of the orbit method that he developed in earlier work with Venkatesh, and an analysis of the geometric side of an appropriate Relative Trace Formula, which facilitates the use of Amplification.