MATHEMATICS

A. N. ANDRIANOV

A GENERALIZATION OF A THEOREM OF M. EICHLER FROM THE THEORY OF QUATERNARY QUADRATIC FORMS

(Presented by Academician I. M. Vinogradov on 3 VI 1961)

Let
\[ F(x_1,\ldots,x_4)=\sum_{1\le i\le j\le 4} a_{ij}x_i x_j \]
be a positive definite quaternary quadratic form with integral rational and relatively prime coefficients \(a_{ij}\). Then
\[ F(x_1,\ldots,x_4)=\frac12 \overline{X}FX, \]
where \(\overline{X}=(x_1,\ldots,x_4)\) and the letter \(F\) denotes the matrix of the form \(F\). Let \(D\) and \(q\) be, respectively, the discriminant and the level of \(F\); \(\mathfrak{C}=(e_1,\ldots,e_4)\) an integral solution of the congruence \(F\mathfrak{C}\equiv 0\pmod q\); \(t=\text{g.c.d.}\left(q,\frac{\overline{\mathfrak{C}}F\mathfrak{C}}{2q}\right)\); \(t_1=\frac qt\); \(\Gamma=\Gamma(q,t_1)\) the group of integral unimodular matrices of the second order
\(\begin{pmatrix}a&b\\ c&d\end{pmatrix}\) such that \(c\equiv 0\pmod q\), \(b\equiv 0\pmod {t_1}\); \(\Gamma_0\) its subgroup consisting of matrices for which \(a\equiv 1\pmod q\). Then the theta series
\[ \vartheta_F(\tau|\mathfrak{C})= \sum_N \exp\left(\pi i t\left(\overline{N}+\frac{\overline{\mathfrak{C}}}{q}\right) F\left(N+\frac{\mathfrak{C}}q\right)\right) = \sum_{n=0}^{\infty} a_F(n)\exp\left(\frac{2\pi i t n}{q_1}\right), \]
where the summation is over all integral 4-dimensional vectors
\(\overline{N}=(n_1,\ldots,n_4)\), and where \(a_F(n)\) is the number of integral solutions of the equation
\[ 2qtn=(qX+\mathfrak{C})F(qX+\mathfrak{C}), \]
is an integral modular form of degree \(q\) and dimension \(-2\) for the group \(\Gamma_0\) \((^2)\). As Hecke showed \((^3)\),
\[ \vartheta_F(\tau|\mathfrak{C})=E_F(\tau|\mathfrak{C})+S_F(\tau|\mathfrak{C}), \]
where \(E_F(\tau|\mathfrak{C})\) is an Eisenstein series, and \(S_F(\tau|\mathfrak{C})\) is a parabolic form. Let \(b_F(n)\) and \(d_F(n)\) be the coefficients of
\[ \exp\left(\frac{2\pi i t n}{q}\right) \]
in the expansions of the forms \(E_F(\tau|\mathfrak{C})\), \(S_F(\tau|\mathfrak{C})\); then
\[ a_F(n)=b_F(n)+d_F(n). \]
The quantity \(b_F(n)\) gives the principal term of the number of solutions of the equation indicated above and is computed without difficulty, while \(d_F(n)\) is the remainder term. In the present note the following is proved.

Theorem. There exists a natural number \(Q\) such that, for all \(n\) relatively prime to \(Q\), the estimate
\[ |d_F(n)|\le c_F \tau(n)\sqrt{n}, \tag{1} \]
holds, where \(c_F\) is a constant depending only on \(F\), and \(\tau(n)\) is the number of divisors of \(n\).

For \(\mathfrak{C}=0\) this estimate was proved by Eichler \((^1)\). The method used by us is analogous to Eichler’s method.

Proof. Denote by \(\Lambda_{2k}\) \((k=1,2,\ldots)\) the set of all parabolic forms \(F(\tau)\) such that, first,
\[ F\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2k}F(\tau), \qquad \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in \Gamma_0 \]

and, secondly, both \(F(\tau)\) itself and the form

\[ (c\tau+d)^{-2k}F\left(\frac{a\tau+b}{c\tau+d}\right) \quad \text{for } \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma \]

have rational, almost integral Fourier coefficients (here and below the coefficients of the Fourier expansion in a neighborhood of the point \(i\infty\) are meant).

Let us note some properties of \(\Lambda_{2k}\):

1) \(\Lambda_{2k}\) is nonempty. Indeed, from the properties of theta series and the series \(E_F(\tau\mid \mathfrak{E})\) \((^{2,3})\) it follows that, for \(\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma\),

\[ S_F\left(\frac{a\tau+b}{c\tau+d}\,\middle|\,\mathfrak{E}\right) = \left(\frac{D}{a}\right)(c\tau+d)^2 S_F(\tau\mid a\mathfrak{E}), \]

where \(\left(\frac{D}{a}\right)\) is the Kronecker symbol, and that \(S_F(\tau\mid \mathfrak{E})\) has rational and almost integral Fourier coefficients. Thus, for \(k=1,2,\ldots\), \(\bigl(S_F(\tau\mid \mathfrak{E})\bigr)^k\in\Lambda_{2k}\).

2) \(\Lambda_{2k}\) is a vector space over the field \(R\) of rational numbers; it is finite-dimensional, since the whole space of modular forms for the group \(\Gamma_0\) of the given dimension \(-2k\) is finite-dimensional.

3) On \(\Lambda_{2k}\) the factor group \(\Gamma/\Gamma_0\), isomorphic to the multiplicative group of residue classes modulo \(q\) relatively prime to \(q\), acts as a group of operators: \(R_a:\Lambda_{2k}\to\Lambda_{2k}\), \((a,q)=1\).

Define on the space \(\Lambda_{2k}\) the Hecke operators \(T(m)\) (see \((^4)\)), putting, for \(F(\tau)\in\Lambda_{2k}\),

\[ F(\tau)\cdot T(m)=m^{2k-1}\sum_{(a,d,b)} F(\tau)\,R_{a'}\begin{pmatrix}a&bt_1\\ 0&d\end{pmatrix}, \]

where the summation is over all triples \((a,d,b)\) satisfying the conditions \(ad=m\), \(a,d>0\), \((a,q)=1\), \(0\le b<d\), where \(a'a\equiv 1\pmod q\), and where, for a form \(G(\tau)\in\Lambda_{2k}\) and an integral matrix \(\begin{pmatrix}l&f\\ p&s\end{pmatrix}\), it is put, for brevity,

\[ (p\tau+s)^{-2k}G\left(\frac{l\tau+f}{p\tau+s}\right) = G(\tau)\begin{pmatrix}l&f\\ p&s\end{pmatrix}. \]

It is known \((^4)\) that

\[ \bigl(F(\tau)\cdot T(m)\bigr)\begin{pmatrix}a&b\\ c&d\end{pmatrix} = \bigl(F(\tau)\cdot R_a\bigr)T(m) \quad \text{for } \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma \]

and, moreover, that together with \(F(\tau)\), \(F(\tau)\cdot T(m)\) for any \(m\) has rational, almost integral Fourier coefficients; thus \(T(m):\Lambda_{2k}\to\Lambda_{2k}\).

Consider the set \(K\) consisting of all functions \(f(\tau)\) of the form \(f(\tau)=\dfrac{F(\tau)}{G(\tau)}\), where \(F(\tau),G(\tau)\in\Lambda_{2k}\), \(k=1,2,\ldots\). It is clear that \(K\) is a field. If \(\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma_0\), then \(f\left(\dfrac{a\tau+b}{c\tau+d}\right)=f(\tau)\), so that \(K\) is a subfield of the field of all modular functions with respect to \(\Gamma_0\), and is the field of algebraic functions of one variable over the field of constants \(R\). Let us note that all functions \(f(\tau)\in K\) have rational, almost integral Fourier coefficients.

Let \(\overline{K}\) be the field obtained from \(K\) by algebraically closing the field of constants. If \((F_i(\tau))\), \(i=1,2,\ldots,g\), is a basis of the space \(\Lambda_2\), then \((F_i(\tau)d\tau)\), \(i=1,2,\ldots,g\), form a basis of the space of integral differentials of the field \(K\), and hence also of \(\overline{K}\). Let \(p\) be a prime, and let the operator \(T(p)\) on \(\Lambda_2\) be given in the basis \((F_i(\tau))\) by the matrix \(T(p)\). Construct a prime multiplier \((^{6a})\) \(\tau_p\) of the field \(\overline{K}\) into itself, which in the basis \(\overline{DU}=(F_i(\tau)d\tau)\) is represented by the matrix \(T(p)\). If \(\overline{K}_0\) is a field abstractly isomorphic to the field \(\overline{K}\), then the prime multipliers of the field \(\overline{K}\) into itself correspond to isomorphisms of the field \(\overline{K}_0\) onto subfields \(\overline{K}_0^{*}\) of finite algebraic extensions \(\overline{K}^{*}\) of the field \(K\) \((^{6a})\). Let the field \(\overline{K}\) consist of functions \(f(\tau)\), and the field \(\overline{K}_0\) of functions \(y(\tau)\), \(y(\tau)\leftrightarrow f(\tau)\);

As \(\overline{K}_0^{*}\) take the field formed by the functions \(f\left(\dfrac{\tau}{p}\right)\), and as the isomorphism \(\overline{K}_0 \to \overline{K}_0^{*}\) take the isomorphism carrying \(y(\tau)\) into \(f\left(\dfrac{\tau}{p}\right)\). Let \(\overline{K}^{*}\) be the composite of \(\overline{K}\) and \(\overline{K}_0^{*}\). It is easy to see that the conjugates of \(f\left(\dfrac{\tau}{p}\right)\) over \(\overline{K}\) will be the functions \(f\left(\dfrac{\tau+t_1 r}{p}\right)\), \(r=0,1,\ldots,p-1\), and \(f(\tau)\cdot R_{p'}\begin{pmatrix}p&0\\0&1\end{pmatrix}\), where \(p'p\equiv 1\pmod q\). Thus \([\overline{K}^{*}:\overline{K}]<\infty\). Let \(\tau_p\) be the multiplier determined by this isomorphism. Put \(\delta u=\left(F_i\left(\dfrac{\tau}{p}\right)d\left(\dfrac{\tau}{p}\right)\right)\), \(i=1,2,\ldots,g\); then to the multiplier \(\tau_p\) there corresponds the matrix \(T^{*}(p)\), determined from relation (66):

\[ \operatorname{Sp}\,\overline{K}^{*}/\overline{K}\,(\delta u)=T^{*}(p)\,du. \]

But since the conjugates of the integral differential \(F_i\left(\dfrac{\tau}{p}\right)d\dfrac{\tau}{p}\) of the field \(\overline{K}^{*}\) over \(\overline{K}\) will be the differentials \(F_i\left(\dfrac{\tau+t_1 r}{p}\right)d\left(\dfrac{\tau+t_1 r}{p}\right)\), \(r=0,1,\ldots,p-1\), and \(F_i(\tau)\,R_{p'}\begin{pmatrix}p&0\\0&1\end{pmatrix}dp\tau\), it is clear that \(T^{*}(p)=T(p)\).

Besides the multipliers \(\tau_p\), the multipliers \(R_a\) for \((a,q)=1\) are naturally defined as the multipliers corresponding to automorphisms \(R_a\) of the field \(K\), defined as follows: if \(f(\tau)=\dfrac{F(\tau)}{G(\tau)}\), \(F(\tau),G(\tau)\in\Lambda_{2k}\), then \(f(\tau)\cdot R_a=\dfrac{F(\tau)\cdot R_a}{G(\tau)\cdot R_a}\). The multipliers \(\tau_p\) and \(R_a\) generate a subring \(M\) of the ring of multipliers of the field \(\overline{K}\) in itself, which (if one multiplies its tensor by \(R\)) is isomorphic to the ring of Hecke operators on \(\Lambda_2\) and is, together with the latter, a commutative semisimple ring of rank \(g\) \((^{4,5})\).

Let \(q\) be a prime number, and let the field \(K\) be given by the equation \(\varphi(x,y)=0\). Then (see \((^7)\)) the congruence \(\varphi(x,y)\equiv0\pmod q\) for almost all \(q\) determines the field \(K(q)\) of algebraic functions of one variable with a field of constants of \(q\) elements, whose genus is equal to the genus of the field \(K\). On the other hand, the field \(K(q)\) can be obtained in the following way: the field \(K\) is generated by two of its functions \(f(\tau)\) and \(g(\tau)\), so that \(K=R(f,g)\), \(\varphi(f,g)=0\). Suppose \(q\) does not enter the denominators of the Fourier coefficients of the functions \(f(\tau)\) and \(g(\tau)\); then for any function \(z(\tau)\in K\) there exists such an exponent \(n\) that \(q^n z(\tau)\) has \(q\)-integral Fourier coefficients, not all divisible by \(q\). Replace them by their residues modulo \(q\) and associate the resulting formal series with the element \(z(\tau)\). Thus a homomorphism of the field \(K\) onto the field of algebraic functions of one variable over the field of \(q\) elements is defined, which, obviously, coincides with \(K(q)\). We shall call admissible those \(q\) for which both reductions are admissible, which, moreover, do not enter the denominators of the coefficients of the forms \(F_i(\tau)\) and modulo which the \(F_i(\tau)\) remain linearly independent. All \(q\), except for a finite number, are admissible. Let \(\overline{K}(q)\) be the field obtained from \(K(q)\) by algebraic closure of the field of constants, and let \(M(q)\) be the image of the ring \(M\) under reduction. \(M\) and \(M(q)\) are semisimple rings (see, respectively, \((^{5,8})\)). The rank of \(M\) is equal to \(g\), the genus of the field \(K\); the rank of \(M(q)\) is also equal to \(g\), for for admissible \(q\) the formal expressions \(F_i(\tau)\,d\tau\) form a basis of the space of integral differentials of the field \(\overline{K}(q)\), and the matrices representing the multipliers of the ring \(M(q)\) in this basis are obtained from the matrices representing the corresponding multipliers of the ring \(M\) by reducing their elements modulo \(q\). Thus the mapping \(M\to M(q)\) is in fact an isomorphism.

Let \(p\) be an admissible prime number; then, considering the behavior of the ideal corresponding to the multiplier \(\tau_p\) in the ring \([\widetilde K,\widetilde K_0]\) (the tensor product over the field of all algebraic numbers), under reduction modulo \(p\), it is easy to see that

\[ \tau_p(p)=R_{p'}(p)\pi^*(p)+\pi(p), \tag{2} \]

where \(\pi(p)\) is the Frobenius multiplier; the asterisk denotes application of the Rosati antiautomorphism \({}^{(6)}\); \(R_{p'}(p)\), \(\tau_p(p)\) are the images of \(R_{p'}\) and \(\tau_p\) under reduction modulo \(p\). It is known \({}^{(9)}\) that the proper numbers of \(\pi(p)\) modulo \(p\) are equal to \(\sqrt p\); thus, in view of the indicated isomorphism \(M \to M(p)\) and relation (2), it follows that the proper numbers of \(\tau_p\), and hence also of \(T(p)\) on \(\Lambda_2\), do not exceed \(2\sqrt p\) in modulus. From this fact and from the general theory of Hecke operators \({}^{(4,5)}\), estimate (1) follows without difficulty, where as \(Q\) one may take the product of all exceptional prime numbers.

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

Received
29 V 1961

CITED LITERATURE

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