TY - JOUR

T1 - Cyclic odd degree base change lifting for unitary groups in three variables

AU - Chan, Ping Shun

AU - Flicker, Yuval Z.

PY - 2009/11

Y1 - 2009/11

N2 - Let F be a number field or a p-adic field of odd residual characteristic. Let E be a quadratic extension of F, and F′ an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (respectively, admissible) representations from the unitary group U(3, E/F) to the unitary group U(3, F′ E/F′). As a consequence, we classify, up to certain restrictions, the packets of U(3, F′ E/F′) which contain irreducible automorphic (respectively, admissible) representations invariant under the action of the Galois group Gal(F′ E/E). We also determine the invariance of individual representations. 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 rigidity, or 'strong multiplicity one', theorem. Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (respectively, admissible) member is Galois-invariant. The restriction that the residual characteristic of the local fields be odd may be removed once the multiplicity one theorem for U(3) is proved to hold unconditionally without restriction on the dyadic places.

AB - Let F be a number field or a p-adic field of odd residual characteristic. Let E be a quadratic extension of F, and F′ an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (respectively, admissible) representations from the unitary group U(3, E/F) to the unitary group U(3, F′ E/F′). As a consequence, we classify, up to certain restrictions, the packets of U(3, F′ E/F′) which contain irreducible automorphic (respectively, admissible) representations invariant under the action of the Galois group Gal(F′ E/E). We also determine the invariance of individual representations. 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 rigidity, or 'strong multiplicity one', theorem. Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (respectively, admissible) member is Galois-invariant. The restriction that the residual characteristic of the local fields be odd may be removed once the multiplicity one theorem for U(3) is proved to hold unconditionally without restriction on the dyadic places.

KW - Automorphic representations

KW - Base change

KW - Langlands functoriality

KW - Trace formula

KW - Unitary groups

UR - http://www.scopus.com/inward/record.url?scp=76449097319&partnerID=8YFLogxK

U2 - 10.1142/S1793042109002687

DO - 10.1142/S1793042109002687

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AN - SCOPUS:76449097319

SN - 1793-0421

VL - 5

SP - 1247

EP - 1309

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 7

ER -