UDC 519.932

MATHEMATICS

A. N. ANDRIANOV

THE SHIMURA CONJECTURE

FOR THE SIEGEL MODULAR GROUP OF GENUS 3

(Presented by Academician Yu. V. Linnik on 16 II 1967)

1. Let \(Z\) be the ring of rational integers;

\[ J_n=\begin{pmatrix} 0&1_n\\ -1_n&0 \end{pmatrix}, \]

where \(1_n\) is the identity matrix of order \(n\); \(\Gamma=\operatorname{sp}(n,Z)\) is the Siegel modular group of genus \(n\). Denote by \(S\) the set of all matrices \(B\) of order \(2n\) with elements from \(Z\), for which \({}^{t}BJ_nB=r(B)J_n\), where \(r(B)\in Z\), \(r(B)>0\). Let \(L\) be the free \(Z\)-module generated by all double classes \(\Gamma B\Gamma\) for \(B\in S\). Introduce multiplication in \(L\) \((^1)\). Let \(L_p\), where \(p\) is a prime number, be the subring of \(L\) generated by those \(\Gamma B\Gamma\) for which \(r(B)\) is a power of \(p\). Then \(L_p\) is a polynomial ring over \(Z\) in \(n+1\) elements

\[ T_{0,n}=T(\underbrace{1,\ldots,1}_{n},\underbrace{p,\ldots,p}_{n}), \]

\[ T_{i,n}=T(\underbrace{1,\ldots,1}_{n-i},\underbrace{p,\ldots,p}_{i}, \underbrace{p^2,\ldots,p^2}_{n-i},\underbrace{p,\ldots,p}_{i})\quad (1\leq i\leq n), \]

where \(T(p^{\alpha_1},\ldots,p^{\alpha_{2n}})\) denotes the double class containing the diagonal matrix with the diagonal indicated in parentheses; these elements are algebraically independent \((^{1,3})\).

Introduce the local Hecke series of the group \(\Gamma\), putting

\[ D_p(s)=\sum_{(\Gamma B\Gamma)\in L_p}(\Gamma B\Gamma)\,r(B)^{-s}. \]

Shimura \((^1)\) conjectured that \(D_p(s)\) is a rational function of \(X=p^{-s}\); more precisely, he assumed that \(D_p(s)=E_n(X)\cdot F_n(X)^{-1}\), where \(E_n(X)\) and \(F_n(X)\) are polynomials in \(X\) of degrees \(2^n-2\) and \(2^n\), respectively. For \(n=1\) the function \(D_p(s)\) was computed by Hecke \((^2)\); for \(n=2\) by Shimura \((^1)\).

The aim of the present note is to compute the function \(D_p(s)\) for \(n=3\). We show that in this case

\[ D_p(s)=\left[\sum_{n=0}^{6}(-1)^{n+1}e(n)X^n\right]\times \left[\sum_{n=0}^{8}(-1)^n f(n)X^n\right]^{-1}, \tag{1} \]

where

\[ e(0)=1,\quad e(1)=0,\quad e(2)=p^2\bigl(T_{2,3}+(p^4+p^2+1)T_{3,3}\bigr), \]

\[ e(3)=p^4(1+p)T_{0,3}T_{3,3},\quad e(4)=p^7\bigl(T_{2,3}T_{3,3}+(p^4+p^2+1)T_{3,3}^2\bigr), \]

\[ e(5)=0,\quad e(6)=-p^{15}T_{3,3}^3;\quad f(0)=1,\quad f(1)=T_{0,3},\quad f(2)=pT_{1,3}+ \]

\[ +p(p^2+1)T_{2,3}+(p^5+p^4+p^3+p)T_{3,3},\quad f(3)=p^3(T_{0,3}T_{2,3}+T_{0,3}T_{3,3}), \]

\[ f(4)=p^6T_{0,3}^2T_{3,3}+p^6T_{2,3}^2-2p^7T_{1,3}T_{3,3} -2p^6(p-1)T_{2,3}T_{3,3}-(p^{12}+2p^{11} \]

\[ +2p^{10}+2p^7-p^6)T_{3,3}^2,\quad f(5)=p^6T_{3,3}f(3),\quad f(6)=p^{12}T_{3,3}^2f(2), \]

\[ f(7)=p^{18}T_{3,3}^3f(1),\quad f(8)=p^{24}T_{3,3}^4. \]

We outline the proof of formula (1).

1. Let \(Q_p\) be the field of \(p\)-adic numbers; \(Z_p\) the ring of integral \(p\)-adic numbers; \(G_p\) the group of all matrices \(A \in GL(2n,Q_p)\) for which
   \({}^{t}AJ_nA=r(A)J_n\), where \(r(A)\in Q_p\); and \(\Gamma_p\) the subgroup of \(G_p\) consisting of matrices \(A\) with coefficients in \(Z_p\), for which \(r(A)\) is a \(p\)-adic unit. Let \(L(G_p,\Gamma_p)\) be the algebra over the field of complex numbers \(\mathbf C\), formed by all complex-valued continuous functions \(\varphi\) on \(G_p\) with compact support, which are constant on the double classes of \(G_p\) with respect to \(\Gamma_p\), where multiplication is defined as convolution:
   \[ \varphi * \psi(A)=\int_{G_p}\varphi(AB^{-1})\psi(B)\,dB; \]
   here \(dB\) is an invariant Haar measure on \(G_p\), normalized by the condition
   \[ \int_{\Gamma_p} dB=1. \]
   Let \(B\in S\) (see paragraph 1) and let \(r(B)\) be a power of the number \(p\). For each such \(B\) denote by \(\overline{\Gamma_pB\Gamma_p}\) the function on \(G_p\) equal to 1 on the set \(\Gamma_pB\Gamma_p\) and to 0 outside this set. From the theory of elementary divisors for the symplectic group \((^3)\) it follows that the mapping
   \[ B\to \overline{\Gamma_pB\Gamma_p}, \]
   extended by linearity to \(L_p\), defines an isomorphism of the ring \(L_p\) onto a subring \(\overline L_p\) of the ring \(L(G_p,\Gamma_p)\). Recall that a function \(\omega\) on \(G_p\) is called zonal spherical if it is constant on the double classes of \(G_p\) with respect to \(\Gamma_p\) and the mapping
   \[ \varphi\to \int_{G_p}\varphi(A)\omega(A^{-1})\,dA=\hat\omega(\varphi) \]
   is a nontrivial homomorphism of the algebra \(L(G_p,\Gamma_p)\) into \(\mathbf C\) \((^{3,4})\).

Using the fact that \(\overline L_p\simeq L_p\) is a polynomial ring over \(Z\), it is easy to see that, in order to compute the function \(D_p(s)\), it suffices to compute its “Fourier transform,” i.e., the function
\[ \hat D_p(s)= \sum_{\overline{\Gamma_pB\Gamma_p}\in \overline L_p} \hat\omega\bigl(\overline{\Gamma_pB\Gamma_p}\bigr)r(B)^{-s} = \]
\[ = \sum_{\overline{\Gamma_pB\Gamma_p}\in \overline L_p} \left(\int_{G_p}\overline{\Gamma_pB\Gamma_p}(A)\omega(A^{-1})\,dA\right)r(B)^{-s} = \]
\[ = \int_{G_p} \left(\sum r(B)^{-s}\overline{\Gamma_pB\Gamma_p}\right)(A)\omega(A^{-1})\,dA = \int_{S_p}\omega(A^{-1})r_0(A)^{-s}\,dA = \xi(s,\omega), \]
where \(\omega\) is an arbitrary zonal spherical function on \(G_p\), \(r_0(A)=p^{\nu_p(r(A))}\), and \(S_p\) is the set of all \(A\in G_p\) with coefficients in \(Z_p\). The function \(\xi(s,\omega)\) is called the \(\zeta\)-function of the group \(G_p\) \((^{3,4})\).

1. We describe the set of zonal spherical functions on \(G_p\) \((^3)\). Let \(H_p\) be the subgroup of all matrices in \(G_p\) of the form
   \[ C=\begin{pmatrix} g & e\\ 0 & {}^{t}g^{-1}p^k \end{pmatrix}_{\}n}^{\}n}, \tag{2} \]
   where \(g\) is a triangular matrix whose entries below the main diagonal are zeros and whose entries on the main diagonal are powers of \(p\). Define a function \(\psi\) on \(H_p\) by putting
   \[ \psi(C)=\lambda_1^{a_1}\cdots \lambda_n^{a_n}\beta^k, \]
   where \(\lambda_1,\ldots,\lambda_n,\beta\) are a fixed set of nonzero complex numbers and \(a_1,\ldots,a_n\) are determined from the condition that the diagonal of \(g\) is equal to \((p^{a_1},\ldots,p^{a_n})\). Since \(\psi(H_p\cap\Gamma_p)=1\) and \(G_p=\Gamma_pH_p=H_p\Gamma_p\) \((^3)\), we can extend \(\psi\) to \(G_p\), putting, for \(A=UC\), \(U\in\Gamma_p\), \(C\in H_p\),
   \(\psi(A)=\psi(C)\). Then the function
   \[ \omega_\psi(A)=\int_{\Gamma_p}\psi(AU)\,dU \]

is a zonal spherical function on \(G_p\), and every zonal spherical function on \(G_p\) can be obtained in the indicated way by a suitable choice of the parameters \(\lambda_1,\ldots,\lambda_n,\beta\) \({}^{(3)}\).

1. Let \(\omega=\omega_\psi\) be a zonal spherical function on \(G_p\) with parameters \(\lambda_1,\ldots,\lambda_n,\beta\). Then it is easy to see that

\[ \xi(s,\omega)=\int_{S_p} r_0^{-s}(A)\left(\int_{\Gamma_p}\psi(A^{-1}U)\,dU\right)dA =\int_{S_p} r_0(A)^{-s}\psi(A^{-1})\,dA. \]

Let \(S_{p,k}\), for \(k=0,1,2,\ldots\), denote the set of all \(A\in S_p\) for which \(r_0(A)=p^k\), and let \(S_{p,k}=\bigcup \alpha_{ik}\Gamma_p\) be a representation of \(S_{p,k}\) as a disjoint union of right classes modulo \(\Gamma_p\). Put \(p^{-s}=X\); then

\[ \xi(s,\omega)=\sum_{k=0}^{\infty}X^k\int_{S_{p,k}}\psi(A^{-1})\,dA =\sum_{k=0}^{\infty}\left(\sum_i \psi(\alpha_{ik}^{-1})\right)X^k. \tag{3} \]

As is known \({}^{(5)}\), as the \(\alpha_{ik}\) one may take a set of matrices of the form (2), where \(g\) runs through all matrices of the form

\[ g= \begin{pmatrix} p^{\alpha_1} & c_{12} & \cdots & c_{1n}\\ 0 & p^{\alpha_2} & & c_{2n}\\ \cdot & & \cdot & \\ \cdot & & & \cdot\\ 0 & 0 & & p^{\alpha_n} \end{pmatrix} \tag{4} \]

with \(\alpha_i\ge 0\), \(c_{i,i+1},\ldots,c_{i,n}\) belonging to the reduced system of residues modulo \(p^{\alpha_i}\) for \(i=1,\ldots,n-1\) (such matrices will be called reduced), and such that the matrix \(p^k g^{-1}\) has integral coefficients, and where, for fixed \(g\), \(e\) runs through a complete set of representatives of the classes into which the set of integral matrices of order \(n\), \(e\), satisfying the condition \(g^t e=e^t g\), is divided with respect to the equivalence relation \(e\sim e_1\leftrightarrow e-e_1=g^t t\) with integral matrix \(t\). Let \(g\) be an integral matrix of order \(n\), \(\det g=p^\nu\), and let \(p^{\nu_i(g)}\), for \(i=1,\ldots,n\), be the greatest common divisor of the minors of order \(i\) of the matrix \(g\), \(\nu_0(g)=0\). Put, for \(i=1,\ldots,n\),
\(\beta_i(g)=\nu_i(g)-\nu_{i-1}(g)\); the numbers \(\beta_i(g)\) are called the elementary divisors of \(g\). One can verify that the number of classes into which the matrices \(e\) are divided with respect to \(g\) is equal to \(p^{n\beta_1(g)+\cdots+\beta_n(g)}\). We also note that the integrality of \(p^k g^{-1}\) is equivalent to the condition \(\beta_n(g)\le k\). Thus,

\[ \sum_i \psi(\alpha_{ik}^{-1}) = \left( \sum_{0\le \beta_1\le\cdots\le \beta_n\le k} p^{\,n\beta_1+(n-1)\beta_2+\cdots+\beta_n} \gamma(\lambda_1,\ldots,\lambda_n;\beta_1,\ldots,\beta_n) \right)\beta^{-k}, \]

where

\[ \gamma(\lambda_1,\ldots,\lambda_n;\beta_1,\ldots,\beta_n) = \sum_g \lambda_1^{-\alpha_1}\cdots \lambda_n^{-\alpha_n}, \]

and here the summation extends over all reduced \(g\) of the form (4) with elementary divisors \(\beta_1,\ldots,\beta_n\).

1. Let now \(F=GL(n,\mathbf Q_p)\), \(U=GL(n,\mathbf Z_p)\). Analogously to item 2, one can define the algebra \(L(F,U)\) and the zonal spherical functions \(\varepsilon=\varepsilon_\psi\) with parameters \(\lambda_1,\ldots,\lambda_n\) on \(F\) \({}^{(4)}\). Let \(\chi(\beta_1,\ldots,\beta_n)\), where the \(\beta_i\) are integers, \(\beta_1\le\cdots\le\beta_n\), be the characteristic function of the double class of \(F\) modulo \(U\) containing the diagonal matrix with diagonal \((p^{\beta_1},\ldots,p^{\beta_n})\). Obviously, \(\chi(\beta_1,\ldots,\beta_n)\in L(F,U)\). Let \(\varepsilon\) be a zonal spherical function on \(F\) with parameters \(\lambda_1,\ldots,\lambda_n\), and let \(\varphi\mapsto \check{\varepsilon}(\varphi)\) be the homomorphism \(L(F,U)\) into \(\mathbf C\) that it determines. Then, using the results of \({}^{(4)}\), it is not difficult to show that

\[ \check{\varepsilon}\bigl(\chi(\beta_1,\ldots,\beta_n)\bigr) = \gamma(\lambda_1,\ldots,\lambda_n;\beta_1,\ldots,\beta_n). \]

Thus, taking into account the results of Sec. 4, we see that in order to compute the series (3) it is sufficient to compute the series

\[ Z_n(X)=\sum_{k=0}^{\infty}\left(\sum_{0\leqslant \beta_1\leqslant\cdots\leqslant \beta_n\leqslant k} \chi(\beta_1,\ldots,\beta_n)p^{n\beta_1+(n-1)\beta_2+\cdots+\beta_n}\right)X^k . \tag{5} \]

1. The ring \(L(F,U)\) is the ring of polynomials over \(C\) in the functions \(\pi_{1,n}=\chi(0,\ldots,0,1),\ldots,\pi_{n,n}=\chi(1,\ldots,1)\) and \(\chi(-1,\ldots,-1)\), the first \(n\) of which are algebraically independent over \(C\) \({}^{(4)}\). However, in order to compute the series (5) we need to know the multiplication table for the functions \(\chi\). In what follows we restrict ourselves to the case \(n=3\). It can be shown that in this case the following multiplication table holds:

\[ \pi_{3,3}\chi(\beta_1,\beta_2,\beta_3) =\chi(\beta_1+1,\beta_2+1,\beta_3+1), \]

\[ \pi_{1,3}\chi(0,e,n) =\chi(0,e,n+1)+\alpha(n-e)\chi(0,e+1,n)+ \]

\[ +(p\alpha(e)+\beta(e,n))\chi(1,e,n), \]

\[ \pi_{2,3}\chi(0,e,n) =\chi(0,e+1,n+1)+\alpha(e)\chi(1,e,n+1)+ \]

\[ +(p\alpha(n-e)+\beta(e+1,n))\chi(1,e+1,n), \]

where \(\alpha(0)=0\), \(\alpha(1)=p+1\), \(\alpha(m)=p\) if \(m>1\), and \(\beta(k,m)=1\) or \(\beta(k,m)=0\) according as \((k,m)=(1,1)\) or \((k,m)\ne(1,1)\).

1. In the notation introduced above, the formula holds

\[ \begin{aligned} Z_3(X)=&\,[1-C(2)X^2+C(3)X^3-C(4)X^4+p^{15}\chi(3,3,3)X^6] \times\\ &\times [(1-X)(1-p\pi_{1,3}X+p^3\pi_{2,3}X^2-p^6\pi_{3,3}X^3)\times\\ &\times(1-p^3\pi_{2,3}X+p^7\pi_{1,3}\pi_{3,3}X^2-p^{12}\pi_{3,3}^2X^3)(1-p^6\pi_{3,3}X)]^{-1}, \end{aligned} \tag{6} \]

where

\[ C(2)=p^6\pi_{3,3}\pi_{1,3}+p^4(1+p+p^2)\pi_{3,3}+p^2\pi_{2,3}, \qquad C(4)=p^5\pi_{3,3}C(2), \]

\[ C(3)=p^4(p+1)(\pi_{3,3}+p\pi_{3,3}\pi_{1,3}+p^3\pi_{3,3}\pi_{2,3}+p^6\pi_{3,3}^{\,2}). \]

Concerning the proof of this formula, we note that all recurrence relations arising in this connection are proved by induction using the rules for multiplying the functions \(\chi\) given in the preceding section.

1. Let us now return to the function \(D_p(s)\). Let \(\varepsilon\) be the same as in Sec. 5. Applying to the left- and right-hand sides of relation (6) the homomorphism \(\hat{\varepsilon}\) of the algebra \(L(F,U)\) and replacing \(X\) by \(X\beta^{-1}\), we obtain an expression for \(\hat{D}_p(s)=\xi(s,\omega)\) in the form of a rational fraction, explicit formulas for the coefficients of whose numerator and denominator are obtained from the easily proved relations

\[ \hat{\varepsilon}(\pi_{1,3}) =\lambda_1^{-1}+p\lambda_2^{-1}+p^2\lambda_3^{-1}=\varphi_1, \]

\[ \hat{\varepsilon}(\pi_{2,3}) =(\lambda_1\lambda_2)^{-1}+p(\lambda_1\lambda_3)^{-1}+p^2(\lambda_2\lambda_3)^{-1}=\varphi_2, \]

\[ \hat{\varepsilon}(\pi_{3,3}) =(\lambda_1\lambda_2\lambda_3)^{-1}=\varphi_3. \]

Formula (1) is now established by eliminating the parameters \(\lambda_1,\lambda_2,\lambda_3,\beta\) from the resulting expression for \(\hat{D}_p(s)\) and the directly verifiable identities

\[ \hat{\omega}(T_{0,3})=\beta^{-1}[1+p\varphi_1+p^3\varphi_2+p^6\varphi_3], \]

\[ \hat{\omega}(T_{1,3})=\beta^2[\varphi_1+(p^2-1)\varphi_2-(p^4+2p^3)\varphi_3+ p^4(p^2-1)\varphi_1\varphi_3+p^8\varphi_2\varphi_3+p^3\varphi_1\varphi_2], \]

\[ \hat{\omega}(T_{2,3})=\beta^{-2}[\varphi_2+(p^3-1)\varphi_3+p^4\varphi_3\varphi_1], \qquad \hat{\omega}(T_{3,3})=\beta^{-2}\varphi_3, \]

where \(\hat{\omega}\) is the homomorphism of the algebra \(L(G_p,\Gamma_p)\) into \(C\), defined with the aid of the function \(\omega\) (see Sec. 2).

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

Received
10 II 1967

REFERENCES

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