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 {{harvs|txt|first=Hiroshi|last=Saito|authorlink=Hiroshi Saito (mathematician)|year1=1975|year2=1975b|year3=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. {{harvtxt|Shintani|1979}} gave a representation theoretic interpretation of Saito's results and used this to generalize them. {{harvtxt|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.
{{harvtxt|Gelbart|1977}}, {{harvtxt|Gérardin |1979}} and {{harvtxt|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 {{harvtxt|Arthur|Clozel|1989}} gives a map from automorphic forms for GLn(F) to automorphic forms for GLn(E) = ResE/FGLn(F). This base change lifting is the special case of Langlands functoriality, corresponding (roughly) to the diagonal embedding of the Langlands dual GLn(C) of GLn to the Langlands dual GLn(C)×...×GLn(C) of ResE/FGLn.
References
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- {{Citation | last1=Saito | first1=Hiroshi | title=Automorphic forms and algebraic extensions of number fields | url=http://www.math.kyoto-u.ac.jp/library/lmku/lmku08.pdf | publisher=Kinokuniya Book-Store Co. Ltd. | location=Tokyo | series=Lectures in mathematics | mr=0406936 | year=1975 | volume=8}}
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- {{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 | 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}}