## Zeta functions of simple algebras.(English)Zbl 0244.12011

Lecture Notes in Mathematics. 260. Berlin-Heidelberg-New York: Springer-Verlag. viii, 188 p. DM 18.00; \$ 5.60 (1972).
The authors extend the theory of Dirichlet series associated to automorphic forms on $$\mathrm{GL}(1)$$ and $$\mathrm{GL}(2)$$ to the multiplicative group $$G$$ of an arbitrary simple algebra $$M$$ over a global field $$F$$. Let $$G_A$$ be the adelic group of $$G$$, $$Z_A$$ the center of $$G_A$$ and $$G_F$$ the subgroups of principal adeles. A continuous function $$\varphi$$ on $$G_F\backslash G_A$$ is said to be cuspidal if the integral $$\displaystyle \int_{U_F\backslash U_A} \varphi(ug)\,dg$$ vanishes, each time $$U$$ is the unipotent radical of a proper $$F$$-parabolic subgroup of $$G$$. Let $$\omega$$ be a character of idele class group of $$F$$. Denote by $$L_0^2(G_F\backslash G_A,\omega)$$ the space of all cuspidal functions $$\varphi$$ on $$G_F\backslash G_A$$ which satisfy the conditions: $$\varphi(ag) = \omega(a)\varphi(g)$$ for $$a\in Z_A$$ and $$g\in G_A$$, and $$\displaystyle \int_{G_FZ_A\backslash G_A} \vert\varphi(g)\vert^2\,dg< +\infty$$. The group $$G_A$$ acts on $$L_0^2$$ by right translations. Let $$K$$ be the standard maximal subgroup of $$G_A$$. A continuous function $$\varphi$$ on $$G_F\backslash G_A$$ is said to be the automorphic form if $$\varphi$$ is $$K$$-finite on the right, the representation of the Hecke algebra $$\mathcal H(G,K)$$ on the space $$\{\varphi * f\mid f\in\mathcal (G,K)\}$$ is admissible, and if $$F$$ is a number field, the function $$\varphi$$ is slowly increasing (in some precise sense). Let $$\mathfrak A_0(G,\omega)$$ be the space of those automorphic forms which lie in $$L_0^2(G_F\backslash G_A,\omega)$$. Any coefficient $$f$$ of unitary representation of $$G_A$$ on $$L_0^2$$ has the form $f(g) = \int_{G_FZ_A\backslash G_A} \varphi(hg)\tilde\varphi(g)\,dh$ with $$\varphi,\tilde\varphi\in L_0^2(G_F\backslash G_A,\omega)$$. If $$\varphi$$ and $$\tilde\varphi$$ are taken in $$\mathfrak A_0(G,\omega)$$ the coefficient is said to be admissible.
The main result (Theorem 13.8, p. 179): If $$f$$ is any admissible coefficient of the representation of $$G_A$$ on the space $$L_0^2(G_F\backslash G_A,\omega)$$, the integral $Z(\Phi,s,f) = \int_{G_A} \Phi(x)f(x) \vert \nu(x)\vert^s\,dx,$ where $$\Phi$$ is a Schwartz-Bruhat function on $$M_A$$, $$dx$$ is a Haar measure on $$G_A$$ and $$\nu(x)$$ is the reduced norm, is absolutely convergent for $$\operatorname{Re} s >n$$ $$(n^2$$ is the rank of $$M$$ over $$F$$). It can be analytically continued as an entire function of $$s$$. It satisfies the functional equation $Z(\Phi,s,f) = Z(\hat\Phi, n-s,\check f),$ where $$\hat\Phi$$ is the Fourier transform of $$\Phi$$ and $$\check f$$ is the coefficient $$g \to f(g^{-1})$$ of the representation of $$G_A$$ on $$L_0^2(G_F\backslash G_A,\omega^{-1})$$. As a corollary the authors obtain the functional equations for corresponding Euler products.
Reviewer: A. N. Andrianov

### MSC:

 11R54 Other algebras and orders, and their zeta and $$L$$-functions 11S45 Algebras and orders, and their zeta functions 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 11R42 Zeta functions and $$L$$-functions of number fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields 20G35 Linear algebraic groups over adèles and other rings and schemes
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