In case ofGLn overp-adic fields, it is known that Shintani base change is well behaved. However, things are not so simple for general reductive groups. In the first part of this paper, we present a counterexample to the existence of quadratic base change descent for some Galois invariant representations. These are representations of type θ10. In the second part, we compute the localL-factor of θ10. Unlike many other supercuspidal representations, we find that theL-factor of θ10 has two poles. Finally, we discuss these two results in relation to the local Langlands correspondence.
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- [AC] J. Arthur and L. Clozel,Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Vol. 120, Annals of Mathematics Studies, Princeton University Press, 1989.Google Scholar
- [HPS] R. Howe and I. Piatetski-Shapiro,A counterexample to the “generalized Ramanujan conjecture”, Proceedings of Symposia in Pure Mathematics33, part 1 (1979), 315–322.Google Scholar
- [La] R. Langlands,Base change for GL(2), Vol. 96, Annals of Mathematics Studies, Princeton University Press, 1980.Google Scholar
- [PS] I. Piatetski-Shapiro,L-functions for GSp(4), Olga Taussky-Todd: In Memoriam. Pacific Journal of Mathematics Special Issue (1997), 259–275.Google Scholar
- [T] J. Tate,Number theoretic background, Proceedings of Symposia in Pure Mathematics33, part 2 (1979), 3–26.Google Scholar