Cambridge Journal of Mathematics

Volume 9 (2021)

Number 2

Congruences of algebraic automorphic forms and supercuspidal representations

Pages: 351 – 429



Jessica Fintzen (University of Cambridge, United Kingdom; and Duke University, Durham, North Carolina, U.S.A.)

Sug Woo Shin (Department of Mathematics, University of California, Berkeley, Calif., U.S.A.; and Korea Institute for Advanced Study, Seoul, South Korea)

Raphaël Beuzart-Plessis (Aix Marseille Université, CNRS, Marseille, France)

Vytautas Paškūnas (Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany)


Congruences between automorphic forms have been an essential tool in number theory since Ramanujan’s discovery of congruences for the $\tau$‑function, for instance in Iwasawa theory and the Langlands program. Over time, several approaches to congruences have been developed via Fourier coefficients, geometry of Shimura varieties, Hida theory, eigenvarieties, cohomology theories, trace formula, and automorphy lifting.

In this paper we construct novel congruences between automorphic forms in quite a general setting using type theory of $p$‑adic groups, generalizing the argument in [Sch18, §7] for certain quaternionic automorphic forms. More precisely, we produce congruences $\operatorname{mod} p^m$ (in the sense of Theorem A below) between arbitrary automorphic forms of general reductive groups $G$ over totally real number fields that are compact modulo center at infinity with automorphic forms that are supercuspidal at $p$ under the assumption that $p$ is larger than the Coxeter number of the absolute Weyl group of $G$. In order to obtain these congruences, we prove various results about supercuspidal types that we expect to be helpful for a wide array of applications beyond those explored in this paper.


supercuspidal representations, types, congruences, automorphic forms

2010 Mathematics Subject Classification

Primary 11F70. Secondary 11F33, 22E50.

The project leading to this publication has received funding from the Excellence Initiative of Aix-Marseille University-A*MIDEX, a French “Investissements d’Avenir” programme.

Received 25 September 2020

Published 7 October 2021