In mathematics, base change lifting is a method of constructing new automorphic form

Automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s 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

Doi–Naganuma lifting

In mathematics, the Doi–Naganuma lifting is a map from elliptic modular forms to Hilbert modular forms of a real quadratic field, introduced by and .It was a precursor of the base change lifting....

 from 1967 was a precursor of the base change lifting. Base change lifting was introduced by for Hilbert modular forms of cyclic totally real fields of prime degree, by comparing the trace of twisted Hecke operator

Hecke operator

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....

s on Hilbert modular forms with the trace of Hecke operators on ordinary modular forms. gave a representation theoretic interpretation of Saito's results and used this to generalize them. 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

Artin conjecture

In mathematics, there are several conjectures made by Emil Artin:* Artin conjecture * Artin's conjecture on primitive roots* The conjecture that finite fields are quasi-algebraically closed; see Chevalley–Warning theorem....

 for tetrahedral and some octahedral 2-dimensional representations of the Galois group.

, and 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 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.