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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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