Abstract
Full Text
A. N. Andrianov
ON THE ANALYTIC ARITHMETIC OF QUADRATIC FORMS WITH AN ODD NUMBER OF VARIABLES IN CONNECTION WITH THE THEORY OF MODULAR FORMS
(Presented by Academician I. M. Vinogradov on 23 II 1962)
Notation.
(f(x_1,\ldots,x_m)=\displaystyle\sum_{1\le i\le j\le m} a_{ij}x_ix_j) is a positive definite quadratic form with integral coefficients and an odd number of variables (m=2k+1); (D) is the discriminant of (f); (q) is the level of (f); (l) is a natural number not divisible by a square different from 1; (r_f(n)) is the number of integral representations of the number (n) by the form (f); (t_0) is the least natural number such that (r_f(lt_0^2)>0); (q_1=4lt_0^2q); (q_2=16l^3t_0^6q); (\Gamma(N)) (respectively (\Gamma(N,N_0)) for (N\equiv0\pmod {N_0})) is the group of integral unimodular matrices of order two (\begin{pmatrix}\alpha&\beta\ \gamma&\delta\end{pmatrix}) such that (\gamma\equiv0\pmod N) (respectively (\gamma\equiv0\pmod N;\ \beta\equiv0\pmod {N_0})); (\left(\dfrac P Q\right)) is the Jacobi symbol;
[
g_l(t)=\sum_{\substack{d\mid t\(d,t_0)=1}} d^{k-1}
\left(\frac{2lD}{d}\right) r_f\left(\frac{lt^2}{d^2}\right);
]
(\tau) is a complex variable.
In this note the following is proved.
Theorem. The function
[
G_l(\tau)=\sum_{t=1}^{\infty} g_l(t)\exp\left(\frac{2\pi i\tau t}{t_0}\right),
]
regular in the upper half-plane (\operatorname{Im}\tau>0), is represented there in the form
[
G_l(\tau)=C_l+F_l(\tau)+\widetilde F_l(\tau),
]
where (C_l) is a constant; (F_l(\tau)), (\widetilde F_l(\tau)) are functions regular for (\operatorname{Im}\tau>0), and
[
F_l\left(\frac{\alpha\tau+\beta}{\gamma\tau+\delta}\right)
=(\gamma\tau+\delta)^{2k}F_l(\tau)
\quad\text{for}\quad
\begin{pmatrix}\alpha&\beta\ \gamma&\delta\end{pmatrix}\in\Gamma(q_2,t_0),
]
[
\widetilde F_l\left(\frac{\alpha\tau+\beta}{\gamma\tau+\delta}\right)
=\left(\frac{2lD}{\alpha}\right)(\gamma\tau+\delta)^k\widetilde F_l(\tau)
\quad\text{for}\quad
\begin{pmatrix}\alpha&\beta\ \gamma&\delta\end{pmatrix}\in\Gamma(q_1).
]
From this theorem, on the basis of the results of the work ((^1)), there follows the following
Corollary. Let (f) be a positive definite ternary form; then there exists a number (Q=Q_{l,f}) such that for all (t) relatively prime to (Q), an asymptotic formula holds for the number of integral representations of the number (lt^2) by the ternary form (f), or, what is the same, for the number of rational points on the ellipsoid (f(x_1,x_2,x_3)=l) with denominator (t):
[
r_f(lt^2)=C_{lf}\sum_{d\mid t} d\prod_{p\mid d}\left(1-\left(\frac{2lD}{p}\right)\frac1p\right)+\alpha_{lf}(t),
\tag{*}
]
where
[
|\alpha_{lf}(t)|<C'_{lf}\tau^2(t)\sqrt t;
]
Here (C_{lf}, C'_{lf}) are certain constants; (\tau(t)) is the number of divisors of the number (t); (p) runs through the prime divisors (d).
Formula (*) may be regarded as a generalization of Gauss’s formulas for the number of integral representations of an integer as a sum of three squares.
Proof of the theorem.
(1^\circ). Put
[
\varphi(x_2,\ldots,x_m)=4a_{11}\sum a_{ij}x_i x_j-(a_{12}x_2+\cdots+a_{1m}x_m)^2 .
]
Denote by (D_\varphi) and (q_\varphi) the discriminant and the level of the form (\varphi). It is easy to see that (D_\varphi=2^{2k-1}a_{11}^{2k-1}D) and that (q_\varphi) divides (q_1=4a_{11}q). Let (b=2lt_0^2), (c=2lt_0), and let (E=(e_2,\ldots,e_m)) be a vector whose components are residue classes modulo (b); let (r_\varphi(n)) and (r_\varphi(n,E)) denote respectively the total number of integral representations of (n) by the form (\varphi), and the number of representations (n=\varphi(x_2,\ldots,x_m)) such that (x_i\equiv e_i\pmod b), (i=2,\ldots,m). Then, since (lt_0^2) is representable by the form (f), we may assume that (a_{11}=lt_0^2), and on the basis of the obvious identity
[
4a_{11}f(x_1,\ldots,x_m)=(2a_{11}x_1+a_{12}x_2+\cdots+a_{1m}x_m)^2+\varphi(x_2,\ldots,x_m)
]
we obtain the following fundamental relation:
[
r_f(lt^2)=\sum_E\sum_{\substack{x=-ct\ x\equiv EH\pmod b}}^{ct}
r_\varphi\bigl((ct)^2-x^2,E\bigr)=\omega_l(t)+\widetilde{\omega}(t),
\tag{1}
]
where the vector (H=(a_{12},\ldots,a_{1m}));
[
\omega_l(t)=\sum_{E\ne0}\sum_{\substack{x=-ct\ x\equiv EH\pmod b}}^{ct}
r_\varphi\bigl((ct)^2-x^2,E\bigr);
\qquad
\widetilde{\omega}(t)=\delta(t')\sum_{x=-t'}^{t'} r_\varphi\bigl((t')^2-x^2\bigr);
]
[
t'=t/t_0;\qquad \delta(t') \text{ is equal to } 1 \text{ if } t' \text{ is an integer, and is equal to } 0 \text{ otherwise.}
]
(2^\circ). Let (\mathfrak M) be some subspace of the space of modular forms for the group (\Gamma(N)), of dimension (k) and character (\chi), closed with respect to the Hecke operators (T(m)), (m=1,2,\ldots) (see ((2))); let (f_1(\tau),\ldots,f_g(\tau)) be a basis of the space (\mathfrak M), and let the operator (T(m)) be given in this basis by the matrix (\lambda(m)). Then ((2))
[
\lambda(m)\lambda(n)=\sum_{d\mid m,n}\lambda!\left(\frac{mn}{d^2}\right)\overline{\chi}(d)d^{k-1},
\tag{2}
]
where (\overline{\chi}) is the character conjugate to (\chi), and (\overline{\chi}(d)=0) if ((d,N)>1). Moreover, if
[
f_i(\tau)=\sum_{m=0}^{\infty} a_i(m)\exp(2\pi i\tau m),
]
then ((2)) there exist (g) constant matrices (B_1,\ldots,B_g), forming a basis of the commutative ring of Hecke operators on (\mathfrak M), such that for (m=1,2,\ldots)
[
\lambda(m)=\sum_{i=1}^{g} a_i(m)B_i.
\tag{3}
]
Let (\alpha_{ij}^k), (i,j,k=1,2,\ldots,g), be the multiplication table for the matrices (B_i), so that
[
B_iB_j=\sum_{k=1}^{g}\alpha_{ij}^k B_k;
]
then from formulas (2) and (3) the following relation is obtained without difficulty:
[
a_s(mn)=\sum_{d\mid m,n}\sum_{i,j=1}^{g}\alpha_{ij}^s
a_i!\left(\frac{m}{d}\right)a_j!\left(\frac{n}{d}\right)\overline{\chi}(d)d^{k-1}\mu(d),
\tag{4}
]
where (m,n>0), and (\mu(d)) is the Möbius function.
3°. Let (E\ne 0). Put (\vartheta_\varphi(\tau\mid E)=\displaystyle\sum_{n=0}^{\infty} r_\varphi(n,E)\exp(2\pi i\tau n)) and denote by (\mathfrak M(\chi)) the space of all modular forms with respect to the group (\Gamma'(q_2)) of character (\chi), of dimension (-k), with zero constant terms of the Fourier expansions in a neighborhood of the point (i\infty). Then from the properties of (\vartheta_\varphi(\tau\mid E)) (see (3)) it follows that
[
\sum_{\substack{r \bmod q_2\ (r,q_2)=1}}
\vartheta_\varphi(\tau\mid rE)\left(\frac{D_\varphi}{r}\right)\overline{\chi}(r)\in \mathfrak M(\chi).
]
Thus, if (f_1^\chi(\tau),\ldots,f_{g_\chi}^\chi(\tau)) is a basis of (\mathfrak M(\chi)) and (\alpha_i(E,\chi)) are the coefficients in the expansion of the last function with respect to the basis (f_i^\chi(\tau)), then
[
\vartheta_\varphi(\tau\mid E)=
\frac{1}{\varphi(q_2)}
\sum_\chi \sum_{i=1}^{g_\chi}
\alpha_i(E\chi)\, f_i^\chi(\tau),
\tag{5}
]
where (\varphi(q_2)) is Euler’s function. To the space (\mathfrak M(\chi)), obviously, the results of the preceding item are applicable; therefore, if we denote by (\alpha_{ij}^{s\chi}), (i,j,s=1,2,\ldots,g_\chi), the multiplication table for the basis of the Hecke operator ring on (\mathfrak M(\chi)), then from relations (4) and (5) it follows that
[
\omega_l(t)=
\frac{1}{\varphi(q_2)}
\sum_\chi
\sum_{i,j,s=1}^{g_\chi}
\alpha_{ij}^{s\chi}
\sum_{d\mid 2ct}
d^{k-1}\mu(d)
\left(\frac{D_\varphi}{d}\right)
b_{ljs}^\chi\left(\frac{2ct}{d}\right),
\tag{6}
]
where
[
b_{ijs}^\chi(2N)=
\sum_{\substack{m+n=2N\ m,n\ge 0}}
a_i^\chi(m)a_j^\chi(n)\,
\delta_s^\chi\left(\frac{n-m}{2}\right)
\quad\text{and}\quad
\delta_s^\chi(r)=
\sum_{E\overline H\equiv r\,(\mathrm{mod}\,b)}
\alpha_s(E,\chi),
]
in the formula for (\delta_s^\chi(0)) the summation is taken over all (E), (E\overline H\equiv0\pmod b), (E\ne0). From relation (6) we obtain:
[
\sum_{d\mid t}
d^{k-1}\left(\frac{D_\varphi}{d}\right)
\omega_l\left(\frac{t}{d}\right)
=
\frac{1}{\varphi(q_2)}
\sum_\chi
\sum_{i,j,s=1}^{g_\chi}
\alpha_{ij}^{s\chi} b_{ljs}^\chi(2ct).
\tag{7}
]
Consider the functions
[
f_i^{\chi,r}(\tau)=
\sum_{n\equiv r\,(\mathrm{mod}\,2b)}
a_i^\chi(n)\exp\left(\frac{2\pi i\tau n}{2b}\right);
]
then, since (2b) divides (q_2), it is easy to prove that for
(\begin{pmatrix} \alpha & \beta \ \gamma & \delta \end{pmatrix}\in\Gamma'(q_2))
[
f_i^{\chi,r}\left(\frac{\alpha\tau+\beta}{\gamma\tau+\delta}\right)
=
\chi(\alpha)\exp\left(\frac{2\pi i r\alpha\beta}{2b}\right)
(\gamma\tau+\delta)^k f_i^{\chi,r\alpha^2}(\tau).
\tag{8}
]
From formula (8) and from the obvious relation
[
B_{ijs}^\chi(\tau)=
\sum_{t=0}^{\infty}
b_{ijs}^\chi(2ct)\exp\left(\frac{2\pi i\tau t}{t_0}\right)
=
\sum_{r+l\equiv0\,(\mathrm{mod}\,2c)}
f_i^{\chi,r}(\tau)f_j^{\chi,l}(\tau)
\delta_s^\chi\left(\frac{l-r}{2}\right)
]
it follows that for
(\begin{pmatrix} \alpha & \beta \ \gamma & \delta \end{pmatrix}\in\Gamma'(q_2,t_0))
[
B_{ijs}^\chi\left(\frac{\alpha\tau+\beta}{\gamma\tau+\delta}\right)
=
(\gamma\tau+\delta)^{2k}B_{ijs}^\chi(\tau).
\tag{9}
]
4°. We now consider
[
\widetilde\omega(t)=
\delta(t')\sum_{x=-t'}^{t'} r_\varphi\bigl((t')^2-x^2\bigr).
]
The series
[
\vartheta_\varphi(\tau)=
\sum_{n=0}^{\infty} r_\varphi(n)\exp(2\pi i\tau n)
]
is a modular form of dimension (-k)
of the character (\chi_0), where (\chi_0(r)=\left(\dfrac{D_\varphi}{r}\right)), with respect to the group (\Gamma'(q_1)) (recall that (q_\varphi) divides (q_1)). Let (\widetilde{\mathfrak M}) be the space of all modular forms with respect to (\Gamma'(q_1)), of dimension (-k), and of character (\chi_0). Let (f_1(\tau),\ldots,f_g(\tau)) be its basis, with (f_1(\tau)=\vartheta_\varphi(\tau)), and let (f_2(\tau),\ldots,f_g(\tau)) have no constant terms in their Fourier expansions (in a neighborhood of the point (i\infty)). We apply to (\widetilde{\mathfrak M}) the results of item (2^\circ). Let the multiplication table of the basis of the Hecke operator ring on (\widetilde{\mathfrak M}) be (\widetilde{\alpha}_{ij}^{\,s}), (i,j,s=1,2,\ldots,g); then, computing (\widetilde{\omega}(t)), we obtain:
[
\widetilde{\omega}(t)
=
\delta(t')
\left(
\sum_{\substack{m+n=2t'\ m,n>0}} r_\varphi(mn)+2
\right)
=
\delta(t')
\left(
\sum_{i,j=1}^{g}\widetilde{\alpha}{ij}^{\,1}
\sum\mu(d)} d^{k-1
\left(\frac{D_\varphi}{d}\right)
b_{ij}!\left(\frac{2t'}{d}\right)
\right.
]
[
\left.
-
2\sum_{i=1}^{g}\widetilde{\alpha}{1i}^{\,1}
\sum\mu(d)} d^{k-1
\left(\frac{D_\varphi}{d}\right)
a_i!\left(\frac{2t'}{d}\right)
+2
\right),
\tag{10}
]
where
[
b_{ij}(n)=
\sum_{\substack{k+s=n\ k,s\ge 0}} a_i(k)a_j(s).
]
From relation (10) we obtain
[
\sum_{d\mid t} d^{k-1}\left(\frac{D_\varphi}{d}\right)
\widetilde{\omega}!\left(\frac{t}{d}\right)
=
]
[
\delta(t')
\left(
\sum_{i,j=1}^{g}\widetilde{\alpha}{ij}^{\,1} b(2t')
-
2\sum_{i=1}^{g}\widetilde{\alpha}{1i}^{\,1} a_i(2t')
+
2\sum\right)} d^{k-1}\left(\frac{D_\varphi}{d
\right).
\tag{11}
]
We note that the series
[
B_{ij}(\tau)=\sum_{t'=0}^{\infty} b_{ij}(2t')\exp(2\pi i\tau t')
]
is a modular form of dimension (-2k), of unit character with respect to the group (\Gamma'(q_1)), and hence also with respect to (\Gamma'(q_2,t_0)); as for the series
[
A_i(\tau)=\sum_{t'=0}^{\infty} a_i(2t')\exp(2\pi i\tau t')
]
and
[
G(\tau)=g_0+\sum_{t'=1}^{\infty}
\left(\sum_{d\mid t'} d^{k-1}\left(\frac{D_\varphi}{d}\right)\right)
\exp(2\pi i\tau t'),
]
they are modular forms with respect to (\Gamma'(q_1)) of dimension (-k) and character (\chi_0(r)=\left(\dfrac{D_\varphi}{r}\right)).
(5^\circ). Since
[
g_l(t)=
\sum_{\substack{d\mid t\ (d,t_0)=1}}
d^{k-1}\left(\frac{2lD}{d}\right)
r_f!\left(\frac{lt^2}{d^2}\right)
=
\sum_{t\mid d}
d^{k-1}\left(\frac{D_\varphi}{d}\right)
\omega_l!\left(\frac{t}{d}\right)
+
\sum_{d\mid t}
d^{k-1}\left(\frac{D_\varphi}{d}\right)
\widetilde{\omega}!\left(\frac{t}{d}\right),
\tag{12}
]
then, applying to formula (12) relations (7), (9), and (11), we obtain the assertion of the theorem.
Remark. On the basis of the computation of concrete examples one may conjecture that, at least for ternary forms, the series (\widetilde{F}_l(\tau)) entering into the formulation of the theorem is identically zero, and (G_l(\tau)), up to a constant, is a modular form of dimension (-2k) with respect to (\Gamma'(q_2,t_0)).
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
19 II 1962
REFERENCES
- A. N. Andrianov, DAN, 141, No. 1, 9 (1961).
- E. Hecke, Math. Ann., a) 114 (1937); b) 114, 316 (1937).
- M. Eichler, Quadratische Formen und orthogonale Gruppen, Berlin, 1952.