UDC 511.61

MATHEMATICS

A. N. ANDRIANOV

CONTINUATION AND A FUNCTIONAL EQUATION FOR ZETA-FUNCTIONS WITH NON-ABELIAN CHARACTERS OF SIMPLE ALGEBRAS OVER NUMBER FIELDS

(Presented by Academician I. M. Vinogradov on 29 XI 1967)

1. Zeta-functions of algebras over number fields were first considered by H. Hey \((^{1,2})\). He defined zeta-functions of algebras with division and proved their analytic continuation and a functional equation. Eichler \((^3)\) generalized these results to zeta-functions with abelian (i.e., one-dimensional) characters. Fujisaki \((^{4,5})\) considered zeta-functions with abelian characters for simple algebras and, reducing them to zeta-functions of the corresponding algebras with division, proved for them continuation and a functional equation.

With the appearance of the theory of Hecke operators \((^6)\), the need arose to study zeta-functions connected with representations of the ring of Hecke operators on spaces of automorphic functions. This formulation of the question leads to the problem of studying zeta-functions with non-abelian characters, where the non-abelian characters that occur are positive-definite zonal spherical functions corresponding to automorphic functions. In general form this problem was posed in lectures of Godement \((^7)\), and there, by the Iwasawa–Tate method, was solved for algebras with division. Tamagawa \((^{8,9})\) developed the theory of the Euler product for non-abelian zeta-functions. Kinoshita \((^{10})\) proved continuation and a functional equation for non-abelian zeta-functions of the full matrix algebra over the field of rational numbers.

We have obtained a general result in this direction: analytic continuation and a functional equation have been proved for zeta-functions with non-abelian characters of an arbitrary simple central algebra over a field of algebraic numbers. We proceed to a detailed formulation of the results.

1. Let \(A\) be a simple central algebra over a finite field of algebraic numbers \(k\); \(G\) the multiplicative group of \(A\); \(\mathfrak D\) a maximal order in \(A\). For each valuation \(\mathfrak p\) of the field \(k\), denote by \(k_{\mathfrak p}\) the completion of \(k\) with respect to \(\mathfrak p\); \(A_{\mathfrak p}=A\otimes_k k_{\mathfrak p}\); \(G_{\mathfrak p}\) the multiplicative group of \(A_{\mathfrak p}\). For non-Archimedean \(\mathfrak p\), denote by \(\mathfrak D_{\mathfrak p}\) the closure of \(\mathfrak D\) in \(A_{\mathfrak p}\), and by \(U_{\mathfrak p}\) the subgroup generated by those \(u_{\mathfrak p}\in G_{\mathfrak p}\) for which \(u_{\mathfrak p}\mathfrak D_{\mathfrak p}=\mathfrak D_{\mathfrak p}\). For Archimedean \(\mathfrak p\) put \(U_{\mathfrak p}=\{u_{\mathfrak p}\in G_{\mathfrak p},\, u u_{\mathfrak p}^{*}=1\}\), where \(a_{\mathfrak p}\mapsto a_{\mathfrak p}^{*}\) is a positive involution of the algebra \(A_{\mathfrak p}\). The subgroup \(U_{\mathfrak p}\subset G_{\mathfrak p}\) is compact for all \(\mathfrak p\); for non-Archimedean \(\mathfrak p\) it is, in addition, open. For \(x_{\mathfrak p}\in A_{\mathfrak p}\) put

\[ V_{\mathfrak p}(x_{\mathfrak p})=\lvert Nx_{\mathfrak p}\rvert_{\mathfrak p}, \]

where \(N\) is the norm in the regular representation of \(A_{\mathfrak p}\) over \(k_{\mathfrak p}\); \(\lvert\ \rvert_{\mathfrak p}\) is the usual norm in the complete field \(k_{\mathfrak p}\), for which the product formula \((^{11})\) holds. Put, for \(x_{\mathfrak p}\in A_{\mathfrak p}\),

\[ \Phi_{\mathfrak p}(x_{\mathfrak p})= \begin{cases} \exp\bigl(-\pi\, \operatorname{Tr}(x_{\mathfrak p}x_{\mathfrak p}^{*})\bigr), & \text{if } \mathfrak p \text{ is Archimedean},\\ \text{the characteristic function of the maximal order } \mathfrak D_{\mathfrak p}\subset A_{\mathfrak p}, & \text{if } \mathfrak p \text{ is non-Archimedean}. \end{cases} \]

where \(\operatorname{Tr}\) is the reduced trace over the field \(\mathbf R\) of real numbers; \(x_{\mathfrak p}\mapsto x_{\mathfrak p}^*\) is a positive involution of \(A_{\mathfrak p}\).

1. Let \(J\) be the group of ideals of the group \(G\), i.e. the restricted direct product \((^{11})\) of the groups \(G_{\mathfrak p}\) with respect to the subgroups \(U_{\mathfrak p}^*\) over all valuations \(\mathfrak p\) of the field \(k\). Denote by \(\Gamma\cong G\) the subgroup of principal ideals and put \(U=\prod_{\mathfrak p} U_{\mathfrak p}\). For \(g=(\ldots g_{\mathfrak p}\ldots)\in J\) put
   \[ V(g)=\prod_{\mathfrak p} V(g_{\mathfrak p});\qquad \Phi(g)=\prod_{\mathfrak p}\Phi_{\mathfrak p}(g_{\mathfrak p}). \]

2. Let \(H\) be a locally compact group; \(V\subset H\) a compact subgroup. Denote by \(L(H,V)\) the algebra of all complex continuous functions \(f\) on \(H\) with compact support which, for all \(v,v'\in V,\ h\in H\), satisfy the condition \(f(vhv')=f(h)\). Multiplication in \(L(H,V)\) is defined as convolution:
   \[ (f*\varphi)(x)=\int_H f(xh^{-1})\varphi(h)\,dh,\qquad x\in H. \]

Recall that a continuous function \(\omega\) on \(H\) is called a (zonal) spherical function with respect to the subgroup \(V\) \((^{8,9,12})\) if for any \(h,h'\in H\)
\[ \int_V \omega(hvh')\,dv=\omega(h)\omega(h'), \]
where \(dv\) is the Haar measure on \(V\), normalized by the condition \(\int_V dv=1\).

1. Let \(\omega\) be a spherical function on \(J\) with respect to \(U\); then
   \[ \omega=\prod_{\mathfrak p}\omega_{\mathfrak p}, \]
   where \(\omega_{\mathfrak p}\) are spherical functions on \(G_{\mathfrak p}\) with respect to \(U_{\mathfrak p}\) \((^8)\). For all \(\mathfrak p\), \(A_{\mathfrak p}\) is a simple algebra over \(k_{\mathfrak p}\), so that \(A_{\mathfrak p}=M_{r_{\mathfrak p}}(D_{\mathfrak p})\), where \(D_{\mathfrak p}\) is a division algebra. The set of spherical functions on \(G_{\mathfrak p}\) (with respect to \(U_{\mathfrak p}\)) is parametrized by \(r_{\mathfrak p}\) complex parameters \((^9)\). We describe this parametrization for Archimedean \(\mathfrak p\). Put
   \[ T_{\mathfrak p}=\{t=(t_{ij})\in G_{\mathfrak p},\ t_{ij}\in D_{\mathfrak p},\ t_{ij}=0\ \text{for}\ i>j,\ t_{ii}\in\mathbf R,\ t_{ii}>0\}. \]
   Then \(G_{\mathfrak p}=U_{\mathfrak p}T_{\mathfrak p}\). Let \((s(\mathfrak p))=(s_1(\mathfrak p),\ldots,s_{r_{\mathfrak p}}(\mathfrak p))\) be arbitrary complex numbers. For \(g_{\mathfrak p}\in G_{\mathfrak p}\), \(g_{\mathfrak p}=u_{\mathfrak p}t_{\mathfrak p}\), where \(u_{\mathfrak p}\in U_{\mathfrak p}\), \(t_{\mathfrak p}\in T_{\mathfrak p}\), put
   \[ \alpha_{(s(\mathfrak p))}(g_{\mathfrak p}) =\alpha_{(s(\mathfrak p))}(t_{\mathfrak p}) =\prod_{i=1}^{r_{\mathfrak p}} t_{ii}^{\nu(-s_i(\mathfrak p)+(i-1))}, \]
   where \(\nu=[D_{\mathfrak p}:\mathbf R]\). Then the function
   \[ \omega_{(s(\mathfrak p))}(g_{\mathfrak p}) =\int_{U_{\mathfrak p}}\alpha_{(s(\mathfrak p))}(g_{\mathfrak p}^{-1}u_{\mathfrak p})\,du_{\mathfrak p} \]
   is a spherical function on \(G_{\mathfrak p}\) with respect to \(U_{\mathfrak p}\), and every spherical function on \(G_{\mathfrak p}\) is obtained in the indicated way by a suitable choice of the parameters \(s_i(\mathfrak p)\). We shall call the parameters \(s_i(\mathfrak p)\) \((i=1,\ldots,r_{\mathfrak p})\) the roots of the spherical function \(\omega_{\mathfrak p}=\omega_{(s(\mathfrak p))}\) and associate to each spherical function \(\omega_{\mathfrak p}\) the polynomial
   \[ P_{\omega_{\mathfrak p}}(z)=\prod_{i=1}^{r_{\mathfrak p}}(r_{\mathfrak p}z-s_i(\mathfrak p))(r_{\mathfrak p}z-s_i(\mathfrak p)-1), \]
   where \(s_i(\mathfrak p)\) are the roots of \(\omega_{\mathfrak p}\).

2. A continuous function \(f\) on \(J\) is called \(\Gamma\)-automorphic if:

I. \(f(ug\gamma)=f(g)\) for all \(u\in U,\ g\in J,\ \gamma\in\Gamma\).

II. For every function \(\varphi\in L(J,U)\) there exists a complex number \(\lambda_\varphi\) such that \(\varphi*f=\lambda_\varphi\cdot f\).

To every nonzero \(\Gamma\)-automorphic function \(f\) on \(J\) there corresponds uniquely a spherical function \(\omega\) on \(J\) (with respect to \(U\)), which for all \(g,\ g'\in J\) satisfies the condition
\[ \int_U f(gug')\,du=\omega(g)f(g') \tag{8}. \]
We shall say that \(\omega\) belongs to \(f\). By the spectrum \(s(\Gamma)\) of the discrete subgroup \(\Gamma\subset J\) we shall mean the set of all spherical functions \(\omega\) on \(J\) with respect to \(U\) that belong to some nonzero \(\Gamma\)-automorphic functions on \(J\), are positive-definite functions on \(J\), and for every \(\xi\) in the center of \(J\) and \(g\in J\) satisfy the relation \(\omega(\xi g)=\omega(g)\).

1. Let \(\Gamma\) be the group of principal ideles, \(\omega\in s(\Gamma)\). The zeta-function of the algebra \(A\) with character \(\omega\) is the function
   \[ \zeta(z,\omega)=\int_J \Phi(g)\omega(g^{-1})V(g)^z\,dg = \]
   \[ = \prod_{\mathfrak p}\int_{G_{\mathfrak p}}\Phi_{\mathfrak p}(g_{\mathfrak p})\omega_{\mathfrak p}(g_{\mathfrak p}^{-1})V_{\mathfrak p}(g_{\mathfrak p})^z\,dg_{\mathfrak p} = \prod_{\mathfrak p}\zeta_{\mathfrak p}(z,\omega_{\mathfrak p}), \]
   where \(dg=\prod_{\mathfrak p}dg_{\mathfrak p}\) is Haar measure on \(J\), and the local measures \(dg_{\mathfrak p}\) on \(G_{\mathfrak p}\), for non-Archimedean \(\mathfrak p\), are normalized by the condition
   \[ \int_{U_{\mathfrak p}} dg_{\mathfrak p}=1. \]
   The function \(\zeta(z,\omega)\) is regular in the domain \(\operatorname{Re} z>1\). The \(\mathfrak p\)-factors \(\zeta_{\mathfrak p}(z,\omega_{\mathfrak p})\) have been computed explicitly by Tamagawa (9); for Archimedean \(\mathfrak p\) they are expressed in terms of the gamma-function, while for non-Archimedean \(\mathfrak p\) they are rational functions of \(p^{-z}\), where \(p\) is the prime number divisible by \(\mathfrak p\). Our main result is formulated as follows:

Main theorem. In the notation and assumptions introduced above, the following assertions hold:

I. The function \(\zeta(z,\omega)\) admits a meromorphic continuation to the whole \(z\)-plane.

II. The function
\[ \zeta(z,\omega)\prod_{\operatorname{arch}} P_{\omega_{\mathfrak p}}(z) \]
is entire.*

III. The function \(\zeta(z,\omega)\) satisfies a functional equation of the form
\[ \zeta(z,\omega)=W(\omega)\Delta^{1/2-z}\zeta(1-z,\bar\omega), \]
where \(W(\omega)\) is a constant depending only on \(\omega\); \(|W(\omega)|=1\); \(\Delta\) is the absolute discriminant of the algebra \(A\).

The proof is based on combining the analysis of weight functions on the Archimedean components with the Poisson formula for the additive group of adeles of the algebra \(A\).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
20 XI 1967

REFERENCES

1. K. Hey, Analytische Zahlentheorie in System hyperkomplexer Zahlen, Diss., Hamburg, 1929.
2. M. Deuring, Algebren, N. Y., 1948.
3. M. Eichler, J. reine u. angew. Math., 179, 227 (1938).
4. G. Fujisaki, J. Fac. Sci. Univ. Tokyo, sect. I, 7, 567 (1958).
5. G. Fujisaki, J. Fac. Sci. Univ. Tokyo, sect. I, 9, 293 (1962).
6. E. Hecke, Math. Ann., 114, 1 (1937).
7. R. Godement, Sem. Bourbaki 1958/1959, Exp. 171, 176.
8. T. Tamagawa, J. Fac. Sci. Univ. Tokyo, sect. I, 8, 363 (1960).
9. T. Tamagawa, Ann. Math., 77, 387 (1963).
10. M. Kinoshita, J. Math. Soc. Japan, 17, No. 4, 374 (1965).
11. J. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta-functions, Thesis, Princeton Univ., 1950.
12. Harish-Chandra, Am. J. Math., 80, 241, 553 (1958).

* This assertion remains valid if, instead of the product over all Archimedean \(\mathfrak p\), one takes an arbitrary one of these polynomials \(P_\omega(z)\).