MATHEMATICS

O. M. FOMENKO

ESTIMATES OF THE PETERSSON INNER PRODUCT WITH AN APPLICATION TO THE THEORY OF QUATERNARY QUADRATIC FORMS

(Presented by Academician I. M. Vinogradov on 5 IV 1963)

Let \(E(x_1,\ldots,x_4)\) be a positive definite quaternary quadratic form with integral rational coefficients having greatest common divisor \(1\); let \(F\) be the matrix of the form; \(D\) the discriminant; \(q\) the level of the form; \(C_1,C_2,\ldots\) absolute positive constants, all of them effective; \(\tau=x+iy\) a complex variable, \(y>0\); \(\Gamma(1)\supset \Gamma_0(q)\supset \Gamma(q)\) the well-known groups of integral unimodular matrices of the second order \({}^{(1)}\); \(\sigma,\sigma_i,\sigma_j,\sigma_l^{(0)}\) matrices from \(\Gamma(1)\); \(t=\mathrm{g.c.d.}\left(q,\dfrac{\bar L F L}{2q}\right)\), where \(\bar L\), as well as \(\bar L^{(1)},\bar L^{(2)},\bar N,\bar G,\bar X\), denotes four-dimensional integral vectors;

\[ \vartheta_F(\tau/L) = \sum_N \exp\left(\pi i t\left(\bar N+\frac{\bar L}{q}\right) F\left(N+\frac{L}{q}\right)\right) = \sum_{n=0}^{\infty}\alpha_F(n,L)\exp\left(\frac{2\pi i ttn}{q}\right) \]

is a theta-series; \(\vartheta_F(\tau/L)=E_F(\tau/L)+S_F(\tau/L)\) \({}^{(2)}\), where \(E_F(\tau/L)\) is an Eisenstein series; \(S_F(\tau/L)\) is an integral parabolic form; \(\varepsilon_F(n,L)\), \(\omega_F(n,L)\) are the coefficients of \(\exp(2\pi i ttn/q)\) in the expansions of these forms. Obviously, \(\alpha_F(n,L)\) is equal to the number of integral solutions of the equation

\[ 2qtn=(q\bar X+\bar L)\,F\,(qX+L). \]

Recall the general fact \({}^{(3)}\) that the space of integral parabolic forms of dimension \(-2k\), belonging to a subgroup \(G\) of the group \(\Gamma\) in (1) of finite index, is a Hilbert space of finite dimension with the Petersson inner product

\[ (f,g)=\iint_{D_G} f(\tau)\overline{g(\tau)}\,y^{2(k-1)}\,dx\,dy, \]

where \(D_G\) is a fundamental domain for the group \(G\). The aim of the present note is to estimate certain inner products of integral parabolic forms connected with representation of a number by the form \(F\), and to obtain from this some arithmetic consequences. Eichler \({}^{(4)}\) gave, for the remainder term \(\omega_F(n,0)\), an estimate unimprovable in the sense of \(n\), showing that

\[ |\omega_F(n,0)|<c_{F,\varepsilon}\,n^{1/2+\varepsilon},\qquad (n,Q)=1. \]

This result was generalized by Shimura \({}^{(5)}\) and A. N. Andrianov \({}^{(6)}\), who showed that

\[ |\omega_F(n,L)|<c_{F,L}\,\tau(n)\sqrt n,\qquad (n,Q)=1, \]

where \(\tau(n)\) is the number of divisors of \(n\); \(Q\) is a certain integer. However, the dependence of the constant \(c_{F,\varepsilon}\) (and \(c_{F,L}\)) on the form was not clarified, which in particular did not make it possible to judge starting from what point

the number \(n\) is representable by the form \(F\). On the basis of estimates of the Petersson inner product of integral parabolic forms we determine this dependence for the case \(L=0\). The general case is investigated analogously.

From Petersson’s results \((^7)\) it follows that the space \(\mathfrak S\left(q,\left(\dfrac{D}{a}\right),q\right)\) of integral parabolic forms of type \(\{\Gamma(q),-2\}\) has a basis

\[ f_i(\tau)=\sum_{n=1}^{\infty}\tau_i(n)e^{2\pi i n\tau}\quad (i=1,2,\ldots,g), \]

consisting of eigenfunctions of all Hecke operators \(T_n\) \((^8)\), where \((n,q)=1\), acting on this space, which is orthogonal, and moreover \(\tau_i(1)=1\) \((i=1,2,\ldots,g)\). It is known \((^3)\) that \(g<C_1q^3\).

Lemma 1. Let \(f(\tau),\varphi(\tau)\) be modular forms of dimension \(-2k\); let \(\Phi\) be some region in the upper half-plane of the \(\tau\)-plane. Then

\[ \iint_{\sigma\Phi} f(\tau)\overline{\varphi(\tau)}\,y^{2(k-1)}\,dx\,dy = \iint_{\Phi} f(\tau)/\sigma\cdot \overline{\varphi(\tau)}/\sigma\, y^{2(k-1)}\,dx\,dy \]

(it is assumed that at least one of the integrals converges).

Lemma 2.

\[ (f_i(\tau),f_i(\tau))>\frac{1}{4\pi e^{4\pi}}q\varphi(q)\quad (i=1,2,\ldots,g). \tag{1} \]

Proof. Let

\[ \Gamma_0(q)=\Gamma(q)\sigma'_1\cup \Gamma(q)\sigma'_2\cup \cdots \cup \Gamma(q)\sigma'_{g_1}, \]

where \(g_1=q\varphi(q)\). Then \((^3)\)

\[ D=\sigma'_1D^{(0)}\cup \sigma'_2D^{(0)}\cup \cdots \cup \sigma'_{g_1}D^{(0)}, \]

where \(D^{(0)},D\) are fundamental regions respectively for the groups \(\Gamma_0(q),\Gamma(q)\). Applying Lemma 1, we have

\[ (f_i(\tau),f_i(\tau)) = \iint_D f_i(\tau)\overline{f_i(\tau)}\,dx\,dy = \sum_{\sigma'_jD^{(0)}}\iint_{D^{(0)}} f_i(\tau)/\sigma'_j\cdot \overline{f_i(\tau)}/\sigma'_j\,dx\,dy. \]

Since \(f_i(\tau)\) are forms of character \(D/a\), the last sum is equal to

\[ q\varphi(q)\iint_{D^{(0)}} f_i(\tau)\overline{f_i(\tau)}\,dx\,dy. \]

Further,

\[ \iint_{D^{(0)}} f_i(\tau)\overline{f_i(\tau)}\,dx\,dy > \iint_{D_0} f_i(\tau)\overline{f_i(\tau)}\,dx\,dy > \frac{1}{4\pi e^{4\pi}}, \]

where \(D_0\) is the fundamental region for the group \(\Gamma(1)\), and one may assume \((^3)\) that it consists of the points \(\tau\) satisfying \(|\tau|>1,\ x<1/2\); \(D^{(0)}\) is chosen so that \(D_0\subset D^{(0)}\).

Remark. Apparently the stronger inequality

\[ (f_i(\tau),f_i(\tau))>C_2q^3 \]

holds, but at present we cannot obtain this result.

Lemma 3.

\[ (S_F(\tau),S_F(\tau))<C_3q^7. \tag{2} \]

Proof. Let

\[ g^2=q^3\prod_{p/q}\left(1-\frac{1}{p^2}\right). \]

It is known \((^3)\) that \(\Gamma(q)\) is a normal divisor of the group \(\Gamma(1)\) of index \(g_2\). Let

\[ \Gamma(1)=\Gamma(q)\sigma_1\cup\Gamma(q)\sigma_2\cup\cdots\cup\Gamma(q)\sigma_{g_2}, \qquad \sigma_i= \begin{pmatrix} \alpha_i & \beta_i\\ \gamma_i & \delta_i \end{pmatrix} \]

\[ (i=1,2,\ldots,g_2). \]

One may assume that \(1 \leqslant \gamma_i \leqslant q\). Accordingly the domain \(D\) is also uniquely determined, since
\[ D=\sigma_1D_0\cup\sigma_2D_0\cup\cdots\cup\sigma_{g_2}D_0. \]
From the system of representatives \(\sigma\) \((i=1,2,\ldots,g_2)\) choose a complete subsystem of elements \(\sigma_i^{(0)}\) that are not equivalent with respect to the equivalence relation \(\sigma_{i_1}^{(0)}\sim\sigma_{i_2}^{(0)}\), which is equivalent to the equality
\[ \sigma_{i_1}^{(0)}=\sigma_{i_2}^{(0)}p_k,\qquad \text{where } p_k\in\Gamma(q) \begin{pmatrix} 1&k\\ 0&1 \end{pmatrix} \quad (k=0,1,\ldots,q-1). \]
The number of elements in such a subsystem is
\[ g_3=q^2\prod_{p/q}\left(1-\frac1{p^2}\right). \]
Shifting \(D_0\) to the right by \(1,2,\ldots,q\) and uniting the resulting domains, we obtain a new domain of width \(q\), which we denote by \(D_1\). Let \(P\) be the part of the upper half-plane of the \(\tau\)-plane bounded by the lines
\[ x=-\frac12,\qquad x=\frac{2q-1}{2},\qquad y=\frac{\sqrt3}{2}. \]
Obviously, \(D_1\subset P\). Let
\[ \sigma_i^{(0)}= \begin{pmatrix} \alpha_i^{(0)}&\beta_i^{(0)}\\ \gamma_i^{(0)}&\delta_i^{(0)} \end{pmatrix} \qquad (i=1,2,\ldots,g_3). \]
It is easy to see that
\[ (S_F(\tau),S_F(\tau))=\sum_{i=1}^{g_3}\iint_{D_1} S_F(\tau)/\sigma_i^{(0)}\cdot \overline{S_F(\tau)/\sigma_i^{(0)}}\,dx\,dy. \]

For the form \(S_F(\tau)\) the transformation formula is valid (for the corresponding formula for \(\vartheta\)-series, see (9)):
\[ S_F(\tau)/\sigma_i^{(0)}= \]
\[ =\frac{1}{(-1)(\gamma_i^{(0)})^2\sqrt D} \sum_{\substack{L\bmod q\\ FL\equiv0\;(\bmod q)}} \exp\left(-\pi i\beta_i^{(0)}\frac{\delta_i^{(0)}\overline{L}FL}{q^2}\right) \varphi(\delta_i^{(0)}L,0)\,S(\tau/L), \]
where
\[ \varphi(\delta_i^{(0)}L,0)= \sum_{\substack{G\bmod(\gamma_i^{(0)}q)\\ G\equiv \delta_i^{(0)}L\;(\operatorname{mod} q)}} \exp\left(\pi i\alpha_i^{(0)}\frac{\overline{G}FG}{\gamma_i^{(0)}q^2}\right). \]

Obviously,
\[ |\varphi(\delta_i^{(0)}L,0)|\leqslant(\gamma_i^{(0)})^4. \]
Now it is easy to obtain the inequality
\[ (S_F(\tau),S_F(\tau))\leqslant \]
\[ \leqslant \sum_{\substack{\delta_i^{(0)}\\ i=1,2,\ldots,g_3}} \frac{(\gamma_i^{(0)})^4}{D} \sum_{\substack{L^{(1)}\bmod q\\ FL^{(1)}\equiv0\;(\bmod q)}} \sum_{\substack{L^{(2)}\bmod q\\ FL^{(2)}\equiv0\;(\bmod q)}} \left|\iint_P S(\tau/L^{(1)})\overline{S(\tau/L^{(2)})}\,dx\,dy\right|. \]

It remains to estimate
\[ \iint_P S(\tau/L^{(1)})\overline{S(\tau/L^{(2)})}\,dx\,dy. \]
We have
\[ S(\tau/L)=\sum_{n=1}^{\infty}\omega_F(n,L)\exp\left(\frac{2\pi i n\tau}{q}\right). \]
But
\[ \omega_F(n,L)=\alpha_F(n,L)-\varepsilon_F(n,L). \]
Proceeding from geometric considerations, one easily obtains the estimate
\[ \alpha_F(n,L)<C_4\frac{n^2t^2}{q^2\sqrt D}. \]
Using the basis of the space of Eisenstein series of dimension \(-2\), which Hecke obtained explicitly (2), it is easy to show that
\[ \varepsilon_F(n,L)<C_5\frac{ntq}{\sqrt D}, \]
whence, for ...

For \(n>C_6q^4\) we obtain
\[ |\omega_F(n,L)|<C_7\frac{n^2t^2}{q^2\sqrt D}. \]
With the aid of this estimate we prove that
\[ \left|\iint_{\dot P} S(\tau/L^{(1)})\overline{S(\tau/L^{(2)})}\,dx\,dy\right|<C_8\frac qD . \tag{3} \]

Using the lemma on the number of solutions of the congruence \(FL\equiv0\pmod q\) \((^{10})\) and (3), we easily obtain (2).

Theorem 1.
\[ |\omega_F(n,0)|<C_9q^4\ln\ln q\,\sqrt n\,\tau(n). \tag{4} \]

Proof. Let
\[ S_F(\tau)=\sum_{i=1}^{g}\alpha_i f_i(\tau). \]
From the results of papers \((^{4-6,\,11})\) it follows that
\[ |\tau_i(n)|\ll \tau(n)\sqrt n,\qquad (n,q)=1. \]
Since
\[ (S_F(\tau),S_F(\tau))=\sum_{i=1}^{g}|\alpha_i|^2(f_i(\tau),f_i(\tau)), \]
the assertion of the theorem follows from Lemmas 2 and 3.

Theorem 2. Let \(n>C_{10}D^{14.01}\) be an odd number and let certain natural congruence conditions be satisfied (see \((^{12})\)). Then \(n\) is representable by the form \(F\), and for the number of representations an asymptotic formula holds.

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

Received
2 IV 1963

REFERENCES

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\(^{3}\) R. C. Gunning, Lectures on Modular Forms, Princeton, 1962.
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