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


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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