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

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

s from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...

to a subgroup. The Doi–Naganuma lifting from 1967 was a precursor of the base change lifting. Base change lifting was introduced by for

Hilbert modular form In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the ''m''-fold product of upper half-planes \mathcal satisfying a certain kind of functional ...

s of cyclic totally real fields of prime degree, by comparing the trace of twisted

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 rep ...

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 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 In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global function fi ...

s, 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''/''F''GL_''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''/''F''GL_''n''.

References

* * * * * * * * *{{Citation , last1=Shintani , first1=Takuro , editor1-last=Borel , editor1-first=Armand , editor1-link=Armand Borel , editor2-last=Casselman , editor2-first=W. , title=Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 , url=https://www.ams.org/publications/online-books/pspum332-index , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

, location=Providence, R.I. , series=Proc. Sympos. Pure Math., XXXIII , isbn=978-0-8218-1437-6 , mr=546611 , year=1979 , chapter=On liftings of holomorphic cusp forms , chapter-url=https://www.ams.org/publications/online-books/pspum332-pspum332-ptIII-5.pdf , pages=97–110 Langlands program