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Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2). (Some arithmetic properties of certain automorphic forms for GL(2)). (French) Zbl 0567.10022
If f is a holomorphic modular form of weight k for \(SL_ 2({\mathbb{Z}})\) with the Fourier expansion \(f(z)=\sum^{\infty}_{n=1}\) \(a_ n \exp (2\pi inz)\) where \(z\in {\mathbb{C}}\) with Im z\(>0\), then, for any automorphism \(\sigma\) of \({\mathbb{C}}\), \(^{\sigma}f\) defined by \(^{\sigma}f(z)=\sum^{\infty}_{n=1}\sigma (a_ n) \exp (2\pi inz)\) is again a cusp form of weight k. If f is a simultaneous eigenform for all Hecke operators and further \(a_ 1=1\), the field \({\mathbb{Q}}(f)\) generated by all the coefficients \(a_ n\) is an algebraic number field; this field plays an important role in the theory of special values of the L-function attached to f. At the very root of the arithmetic theory of modular forms lie these results and their far-reaching generalizations proved with the help of the Eichler-Shimura isomorphism theorem. Adèlic and group-theoretic techniques have taken these further to an entirely new framework, involving the cohomology of arithmetic groups, questions of rationality and special values of associated L-functions.
A systematic account of the corresponding results for cuspidal automorphic representations of \(GL_ 2\) over a number field F is given in the first section while the next section gives a parallel study for admissible irreducible representations of the multiplicative group \(M^{\times}\) of a quaternion algebra M over F. (The field of rationality \({\mathbb{Q}}(\pi)\) for a representaion \(\pi\) of \(M^{\times}\) as above or for a special or cuspidal representation \(\pi\) of \(PGL_ 2(F)\) is even a cyclotomic field.) By analogy with a result of Shimura, the third section is concerned with exhibiting a subspace of ’arithmetic’ automorphic forms in the representation space E for a given representation \(\pi\) as above, say, of \(PGL_ 2(F)\); these ’arithmetic’ forms f (which more or less generate E) are characterized essentially by the integral of f along a maximal subtorus of \(PGL_ 2\) to \({\mathbb{Q}}(\pi)\), the field of rationality (upto an explicit multiplicative factor).
Reviewer: S.Raghavan

MSC:
11F12 Automorphic forms, one variable
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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References:
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