Current Developments in Mathematics

Volume 2021

Reciprocity and symmetric power functoriality

Pages: 95 – 162



Jack A. Thorne (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)


Symmetric power functoriality is the one of the first interesting cases of the Langlands functoriality conjectures, which predict the existence of liftings of automorphic forms from one reductive group to another. These conjectures are closely tied, through the theory of $L$-functions, to questions in number theory of independent interest, most famously the Sato–Tate conjecture.

In this article we first give an introduction to these $L$-functions and their connection with the Langlands programme, before giving a guide to our proof, with James Newton, of symmetric power functoriality for holomorphic modular forms.

The author’s work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714405).

Published 2 August 2023