Cyclic odd degree base change lifting for unitary groups in three variables

TL;DR

This paper establishes a base-change lifting for automorphic representations from U(3,E/F) to U(3,F'E/F') over odd degree cyclic extensions, classifies invariant packets, and explores new phenomena in automorphic representation theory.

Contribution

It introduces the first base-change lifting for groups with non-singleton packets and no strong multiplicity one, revealing novel invariant packet behaviors.

Findings

Classified invariant packets containing Galois-invariant automorphic representations.

Established local twisted character identities for the base-change lifting.

Identified new phenomena where not all automorphic members in invariant packets are Galois-invariant.

Abstract

Let E/F be a quadratic number (resp. p-adic) field extension, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (resp. admissible) representations from the unitary group U(3,E/F) associated with E/F to the unitary group U(3,F'E/F'). As a consequence, we classify the invariant packets of U(3,F'E/F'), namely those which contain (irreducible) automorphic (resp. admissible) representations which are invariant under the action of the Galois group Gal(F'E/E). To do this we use the trace formula technique, and well-known results on the base-change lifting from U(3,E/F) to GL(3,E) and on the base-change lifting for the general linear groups. We also determine the invariance of individual representations, using Howe correspondence. This work is the first study of base change into an algebraic group whose packets are not all singletons, and which does not satisfy the "strong multiplicity one theorem." Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (resp. admissible) member is Galois-invariant. We also obtain local twisted character identities with respect to this base-change lifting.