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Base change for $$\mathrm{GL}(2)$$. (English) Zbl 0444.22007
Annals of Mathematics Studies, 96. Princeton, New Jersey: Princeton University Press. Tokyo: University of Tokyo Press. VII, 237 p. hbk: $22.00, pbk:$ 8.75 (1980).
This volume reproduces Langlands’ famous mimeographed notes of 1975, in which he solved the base change problem for $$\mathrm{GL}(2)$$. Let $$F$$ be a global field of characteristic zero, $$E/F$$ a cyclic extension of prime degree, $$A_F$$, $$A_E$$ the respective adèles. The question is to relate automorphic representations of $$\mathrm{GL}(2, A_E)$$ to irreducible automorphic representations of $$\mathrm{GL}(2, A_F)$$. Langlands’ functoriality principle suggests that there should be a way to do this that is “functorial” – preserves $$L$$-functions, etc., and this book checks that it is indeed so. If we think for a moment about $$\mathrm{GL}(1)$$, the corresponding question is how to turn a grössencharakter for $$F$$ into a grössencharakter for $$E$$, and the answer is easy: just compose with the norm $$A_E^\times \to A_F^\times$$.
The difficulty for $$\mathrm{GL}(2)$$ is that because it is not commutative, there is no norm homomorphism. Much of the actual work is done locally: let $$F_v$$ be a completion of $$F$$, $$E_w$$ a completion of $$E$$ lying over $$F_v$$; we need to compare representations of $$\mathrm{GL}(2, E_w)$$ and $$\mathrm{GL}(2, F_v)$$. The representations of $$\mathrm{GL}(2, E_w)$$ that enter into the discussion will be those that are invariant under $$\mathrm{Gal}(E_w/F_v)$$, and can thus be extended to representations of the semi-direct product $$\mathrm{GL}(2, E_w)\times \mathrm{Gal}(E_w/F_v)$$. Each such representation has a trace, which is represented by a function on the group, known as its character. Letting $$\sigma$$ be a generator of $$\mathrm{Gal}(E_w/F_v)$$, we consider the restriction of the character to the coset $$\mathrm{GL}(2, E_w)\times \{\sigma\}$$, which we identify with $$\mathrm{GL}(2, E_w)$$. The “twisted” character so obtained on $$\mathrm{GL}(2, E_w)$$ is invariant under “$$\sigma$$-conjugacy” $$(x,y$$ are $$\sigma$$-conjugate if for some $$g$$, $$x= g^{-1}yg^\sigma)$$.
To compare automorphic representations of $$\mathrm{GL}(2,A_F)$$ and the appropriate representations of $$\mathrm{GL}(2,A_E)$$, one is led to a comparison of the trace formula for $$\mathrm{GL}(2,A_F)$$ and a “$$\sigma$$-twisted trace formula” for $$\mathrm{GL}(2,A_E)$$, that is a trace formula in which conjugacy is replaced by $$\sigma$$-conjugacy, and orbital integrals (integrals over conjugacy classes) are replaced by integrals over $$\sigma$$-conjugacy classes. Although it is not a homomorphism there is a norm mapping from $$\mathrm{GL}(2, E_w)$$ to $$\mathrm{GL}(2, F_v)$$, and it takes $$\sigma$$-conjugacy classes in $$\mathrm{GL}(2, E_w)$$ to conjugacy classes in $$\mathrm{GL}(2, F_v)$$. If $$\pi$$ is an irreducible admissible representation of $$\mathrm{GL}(2, F_v)$$, then the composition of its character with the norm gives a $$\sigma$$-conjugate-invariant function on $$\mathrm{GL}(2, E_w)$$. If this function is the “twisted” character of a representation $$\Pi$$ of $$\mathrm{GL}(2, E_w)$$, as described above, then $$\Pi$$ is called the “lifting” of $$\pi$$ (the precise definition of lifting is slightly more complicated).
After a considerable amount of analysis of the local situation, it is possible to turn to the global question, the comparison of the trace formula for $$\mathrm{GL}(2,A_F)$$ with the twisted trace formula for $$\mathrm{GL}(2,A_E)$$. Local information shows that most parts of the two formulae are equal, and then some additional hard work shows the remaining parts are in fact also equal. This proves the existence of global liftings for suitable representations of $$\mathrm{GL}(2,A_F)$$.
Finally, all this can be used to prove certain cases of Artin’s Conjecture (that certain $$L$$-functions are entire). The basic idea is not hard to state (though it is much more difficult to put into practice). To show that a certain $$L$$-function is entire one wants to show that it is the $$L$$-function attached to a cuspidal representation of $$\mathrm{GL}(2,A_F)$$. The theory of base change gives a new strength to this approach. Sometimes it is difficult to construct the representation of $$\mathrm{GL}(2,A_F)$$ directly. It may be easier first to prove the existence of a suitable representation of $$\mathrm{GL}(2,A_E)$$, and then show that it is the lifting of the required representation of $$\mathrm{GL}(2,A_F)$$.
This volume reproduces the original notes, with minor corrections, but there are also two helpful additions. One is a chapter which summarizes the entire work, and the other is a chapter describing clearly and in considerable detail the application to Artin’s Conjecture. The original notes, famous for solving certain cases of the conjecture, hardly even mentioned it, so this section is most welcome.
No review would be complete without mentioning the excellent summary of this subject written by P. Gérardin and J. P. Labesse [Proc. Symp. Pure Math. 33, No. 2, 115–133 (1979; Zbl 0412.10018)].
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 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 11-02 Research exposition (monographs, survey articles) pertaining to number theory 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11R39 Langlands-Weil conjectures, nonabelian class field theory 11F70 Representation-theoretic methods; automorphic representations over local and global fields