Base change lifting

In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galois group to a subgroup.

The Doi–Naganuma lifting from 1967 was a precursor of the base change lifting. Base change lifting was introduced by Hiroshi Saito (1975, 1975b, 1979) for Hilbert modular forms of cyclic totally real fields of prime degree, by comparing the trace of twisted Hecke operators on Hilbert modular forms with the trace of Hecke operators on ordinary modular forms. Shintani (1979) gave a representation theoretic interpretation of Saito's results and used this to generalize them. Langlands (1980) extended the base change lifting to more general automorphic forms and showed how to use the base change lifting for GL₂ to prove the Artin conjecture for tetrahedral and some octahedral 2-dimensional representations of the Galois group.

Gelbart (1977), Gérardin (1979) and Gérardin & Labesse (1979) gave expositions of the base change lifting for GL₂ and its applications to the Artin conjecture.

Properties

If E/F is a finite cyclic Galois extension of global fields, then the base change lifting of Arthur & Clozel (1989) gives a map from automorphic forms for GL_n(F) to automorphic forms for GL_n(E) = Res_E/FGL_n(F). This base change lifting is the special case of Langlands functoriality, corresponding (roughly) to the diagonal embedding of the Langlands dual GL_n(C) of GL_n to the Langlands dual GL_n(C)×...×GL_n(C) of Res_E/FGL_n.

References

• Arthur, James; Clozel, Laurent (1989), Simple algebras, base change, and the advanced theory of the trace formula (PDF), Annals of Mathematics Studies, 120, Princeton University Press, ISBN 978-0-691-08517-3, MR 1007299, archived from the original (PDF) on 2011-09-06
• Gelbart, Stephen (1977), "Automorphic forms and Artin's conjecture", Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn., Bonn, 1976), Lecture Notes in Math., 627, Berlin, New York: Springer-Verlag, pp. 241–276, doi:10.1007/BFb0065304, MR 0568306
• Gérardin, Paul (1979), "Changement du corps de base pour les représentations de GL(2) [d'après R. P. Langlands, H. Saito, et T. Shintani]", Séminaire Bourbaki, 30e année (1977/78), Lecture Notes in Math., 710, Berlin, New York: Springer-Verlag, pp. 65–88, doi:10.1007/BFb0069973, MR 0554215
• Gérardin, P.; Labesse, Jean-Pierre (1979), "The solution of a base change problem for GL(2) (following Langlands, Saito, Shintani)", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 115–133, ISBN 978-0-8218-1435-2, MR 0546613
• Langlands, Robert P. (1980), Base change for GL(2), Annals of Mathematics Studies, 96, Princeton University Press, ISBN 978-0-691-08263-9, MR 0574808
• Saito, Hiroshi (1975), Automorphic forms and algebraic extensions of number fields (PDF), Lectures in mathematics, 8, Tokyo: Kinokuniya Book-Store Co. Ltd., MR 0406936
• Saito, Hiroshi (1975b), "Automorphic forms and algebraic extensions of number fields", Proceedings of the Japan Academy, 51 (4): 229–233, doi:10.3792/pja/1195518624, ISSN 0021-4280, MR 0384703
• Saito, Hiroshi (1979), "Automorphic forms and algebraic extensions of number fields. II", Journal of Mathematics of Kyoto University, 19 (1): 105–123, ISSN 0023-608X, MR 0527398
• Shintani, Takuro (1979), "On liftings of holomorphic cusp forms" (PDF), in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 97–110, ISBN 978-0-8218-1437-6, MR 0546611