Abstract
Full Text
A. V. MALYSHEV
ON GAUSS SUMS AND KLOOSTERMAN SUMS
(Presented by Academician I. M. Vinogradov on 9 IV 1960)
Let
[
f=f(x_1,\ldots,x_n)=\sum_{\alpha,\beta=1}^{n} a_{\alpha\beta}x_\alpha x_\beta
]
be an integral quadratic form, so that (a_{\alpha\alpha}) ((\alpha=1,\ldots,n)) and (a_{\alpha\beta}+a_{\beta\alpha}=2a_{\alpha\beta}) ((\alpha,\beta=1,\ldots,n;\ \alpha\ne\beta)) are integers; let
[
l=l(x_1,\ldots,x_n)=\sum_{\alpha=1}^{n} c_\alpha x_\alpha
]
be an integral linear form; let (q) and (g) be positive integers, and let (b_1,\ldots,b_n) be integers. The sum
[
S_{g;\,b_1,\ldots,b_n}(f,l;q)=
\frac{1}{g^n}\sum_{x_1,\ldots,x_n=0}^{q-1}
\exp\left[2\pi i\,
\frac{f(gx_1+b_1,\ldots,gx_n+b_n)+l(gx_1+b_1,\ldots,gx_n+b_n)}{q}
\right]
\tag{1}
]
will be called an inhomogeneous multiple Gauss sum modulo (g). These sums find application in the arithmetic of quadratic forms. If (g=1,\ b_1=\cdots=b_n=0), then we have an inhomogeneous multiple Gauss sum over a complete system of residues
[
S(f,l;q)=S_{1,0,\ldots,0}(f,l;q)=
\sum_{x_1,\ldots,x_n=0}^{q-1}
\exp\left[2\pi i\,
\frac{f(x_1,\ldots,x_n)+l(x_1,\ldots,x_n)}{q}
\right].
\tag{2}
]
If, moreover, (l=0), then we have the homogeneous multiple Gauss sum
[
S(f;q)=S(f,0;q)=
\sum_{x_1,\ldots,x_n=0}^{q-1}
\exp\left[2\pi i\,
\frac{f(x_1,\ldots,x_n)}{q}
\right].
\tag{3}
]
If (n=1,\ f=ax^2), then (3) turns into the ordinary Gauss sum
[
S(a;q)=S(ax^2;q).
]
The sums (S(a;q)) were already studied by Gauss ((^1)). The original theory of the sums (3), given by Weber ((^2)), was later substantially simplified by Minkowski ((^3)). Weber ((^2)) and Kloosterman ((^4)) considered special cases of the sums (2). Here we give formulas that make it possible to compute Gauss sums in their most general form (1).
Theorem 1. 1) If (g) is a complete system of residues ((x_1,\ldots,x_n)\pmod q), ((b'_1,\ldots,b'_n)\equiv(b_1,\ldots,b_n)\pmod{\mathrm{g.c.d.}\,(g,q)}), then
[
S_{g;\,b_1,\ldots,b_n}(f,l;q)=
]
[
=\frac{1}{g^n}
\sum_{(x_1,\ldots,x_n)\in B}
\exp\left[
2\pi i\,
\frac{
f(gx_1+b'_1,\ldots,gx_n+b'_n)+l(gx_1+b'_1,\ldots,gx_n+b'_n)
}{q}
\right].
]
2) If (g=g_1g_2), (\mathrm{g.c.d.}\,(g_2,q_1)=1), then
[
S_{g;\,b_1,\ldots,b_n}(f,l;q)=
\frac{1}{g_2^n}S_{g_1;\,b_1,\ldots,b_n}(f,l;q).
]
3) If (\gcd(a,q)=1), then
[
S_{g;\,b_1,\ldots,b_n}(a^2f,al;q)=S_{g;\,ab_1,\ldots,ab_n}(f,l;q).
]
4) If (d\mid q), (d\mid f), then
[
S_{g;\,b_1,\ldots,b_n}(f,l;q)=
\begin{cases}
d^n S_{g;\,b_1,\ldots,b_n}\left(\dfrac{f}{d},\dfrac{l}{d};\dfrac{q}{d}\right), & \text{if } d\mid l,\[6pt]
0, & \text{if } d\nmid gl.
\end{cases}
]
5) Let (q=q_1q_2\cdots q_k), where the positive integers (q_1,\ldots,q_k) are pairwise coprime; (Q_1=q/q_1,\ldots,Q_k=q/q_k). Then(^*)
[
S_{g;\,b_1,\ldots,b_n}(f,l;q)
=
g^{n(k-1)}
\prod_{\alpha=1}^{k}
S_{g;\,b_1,\ldots,b_n}!\left(
Q_\alpha^{-1\,(\operatorname{mod} q_\alpha)}f,
Q_\alpha^{-1\,(\operatorname{mod} q_\alpha)}l;
q_\alpha
\right).
]
6) If, under an integral substitution whose determinant is relatively prime to (q), the forms (f,l) and the vector ((b_1,\ldots,b_n)) are transformed respectively into the forms (f',l') and the vector ((b'_1,\ldots,b'_n)), then
[
S_{g;\,b_1,\ldots,b_n}(f,l;q)
=
S_{g;\,b'_1,\ldots,b'_n}(f',l';q).
]
7) Let
[
f(x_1,\ldots,x_n)=\sum_{\alpha=1}^{k} f_\alpha(x_{\alpha_1},\ldots,x_{\alpha n_\alpha}),\quad
l(x_1,\ldots,x_n)=\sum_{\alpha=1}^{k} l_\alpha(x_{\alpha_1},\ldots,x_{\alpha n_\alpha}),
]
where ({1,\ldots,n}) is the union of pairwise disjoint sets of indices ({(\alpha 1),\ldots,(\alpha n_\alpha)}) ((\alpha=1,\ldots,k)). Then
[
S_{g;\,b_1,\ldots,b_n}(f,l;q)
=
\prod_{\alpha=1}^{k}
S_{g;\,b_1,\ldots,b_{\alpha n_\alpha}}(f_\alpha,l_\alpha;q).
]
The following known proposition holds ((^5)).
Remark. Let
[
q=\prod_{p\mid q} p^{t(p)}
]
be the factorization of the positive integer (q) into distinct prime factors (p). Every integral quadratic form (f) is equivalent to a form (f_1) for which, for every prime number (p\mid q),
[
f_1\equiv \varphi^{(p)}
=
\sum_{\alpha=1}^{s(p)}
p^{e_\alpha(p)}\varphi_\alpha^{(p)}
\pmod{p^{t(p)}},
\tag{4}
]
where
[
-1\le e_1(p)<e_2(p)<\cdots<e_{s(p)}(p)<t(p);
]
(\varphi_1^{(p)},\ldots,\varphi_{s(p)}^{(p)}) are forms with integral matrices whose variables are pairwise disjoint; the determinants (d_1(p),\ldots,d_{s(p)}(p)) of these forms are relatively prime to (p).
We additionally define: (n_\alpha(p)) to be the number of variables of the form (\varphi_\alpha^{(p)}) ((\alpha=1,\ldots,s(p)));
[
n(p)=\sum_{\alpha=1}^{s(p)} n_\alpha p,\quad 0\le n(p)\le n;
]
(\sigma(\varphi_\alpha^{(2)})=1), if (2\nmid \varphi_\alpha^{(2)});
[
\sigma(\varphi_\alpha^{(2)})=2,\quad \text{if } 2\mid \varphi_\alpha^{(2)};
]
[
\varphi_{s(p)+1}^{(p)}=f_1-\varphi^{(p)};\quad
n_{s(p)+1}(p)=n-n(p);\quad
e_{s(p)+1}=t(p).
]
Theorem 2. Let
[
q=\prod_{p\mid q} p^{t(p)}=2^{t(2)}q_1
]
be the factorization of the positive integer (q) into distinct prime factors (p); (q_1) is odd; (t(2)\ge0); (t(p)>0), if (p\ne2). Let the integral quadratic form (f) in (n) variables, for every (p\mid q), be equivalent ((\bmod\, p^{t(p)})) to a form (\varphi^{(p)}) of the form (4). Then (S(f;q)=0), if (\sigma(\varphi_{s(2)}^{(2)})=1) and (e_{s(2)}(2)+1=t(2)). In the opposite case
[
S(f;q)=(-1)^{
\frac{q_1-1}{2}
\sum_{\alpha=1}^{s(2)}
\sigma!\left(\varphi_\alpha^{(2)}\right)
\frac{(-1)^{n_\alpha(2)(n_\alpha(2)+1)/2}d_\alpha(2)-1}{2}
}
\times
]
(^*) By (a^{-1}(\bmod m)) we denote the number (a_1(\bmod m)) for which (a_1a\equiv1\pmod m).
[
\begin{gathered}
\times i^{\left(\frac{q_1-1}{2}\right)^2
\sum_{\alpha=1}^{s(2)} n_\alpha(2)\, s(2)}
\prod_{\alpha=1}^{s(2)}
{-c_2(\varphi_\alpha^{(2)})}^{\sigma(\varphi_\alpha^{(2)})}
\times
\
\times
\prod_{\alpha=1}^{s(2)}
\left(\frac{2}{d_\alpha(2)}\right)^{t(2)-e_\alpha(2)-\sigma(\varphi_\alpha^{(2)})+1}
\times
\prod_{p\mid q_1}\prod_{\alpha=1}^{s(p)}
\left(\frac{d_\alpha(p)}{p}\right)^{t(p)-e_\alpha(p)}
\times
\
\times i^{
\sum_{\alpha=1}^{s(2)}
\sigma(\varphi_\alpha^{(2)})^2
\left(\frac{d_\alpha(2)-1}{2}\right)^2}
\times
\prod_{p\mid q_1}
\left(\frac{2}{p}\right)^{
t(2)\sum_{\alpha=1}^{s(p)} n_\alpha(p)(t(p)-e_\alpha(p))
+
t(p)\sum_{\alpha=1}^{s(2)} n_\alpha(2)(t(2)-e_\alpha(2))}
\times
\
\times
\prod_{\substack{p\mid q_1,\ p'\mid q_1\ p\ne p'}}
\left(\frac{p'}{p}\right)^{
t(p')\sum_{\alpha=1}^{s(p)} n_\alpha(p)(t(p)-e_\alpha(p))}
\times
\
\times i^{
\sum_{p\mid q_1}
\left(\frac{p-1}{2}\right)^2
\sum_{\alpha=1}^{s(p)} n_\alpha(p)(t(p)-e_\alpha(p))^2}
\times
\left(\frac{1+i}{\sqrt2}\right)^{
\sum_{\alpha=1}^{s(2)} n_\alpha(2)(2-\sigma(\varphi_\alpha^{(2)}))}
\times
\
\times
(2q)^{\frac n2}
\times
2^{
\frac12(n-n(2))(t(2)-1)
+
\frac12\sum_{\alpha=1}^{s(2)} n_\alpha(2)e_\alpha(2)}
\times
\
\times
\prod_{p\mid q_1}
p^{
\frac12(n-n(p))t(p)
+
\frac12\sum_{\alpha=1}^{s(p)} n_\alpha(p)e_\alpha(p)}
,
\end{gathered}
\tag{5}
]
where (c_2(\varphi)) is the Hasse invariant ((^5)).
Theorem 3. Let
[
q=\prod_{p\mid q}p^{t(p)}=2^{t(2)}q_1;\qquad t(2)\geq 0;\qquad t(p)>0,
]
if (p) is an odd prime; (Q(p)=\dfrac{q}{p^{t(p)}}). Let the integral quadratic form
(f=f(x_1,\ldots,x_n)=f_1) satisfy congruence (4) for all primes (p\mid q). Let
[
l=l(x_1,\ldots,x_n)=\sum_{\beta=1}^{n}c_\beta x_\beta
]
be an integral linear form;
[
l=\sum_{\alpha=1}^{s(p)+1}l_\alpha^{(p)},
]
where the variables of the forms (l_\alpha^{(p)}) coincide with the variables of the quadratic form (\varphi_\alpha^{(p)}). Let
[
\tau_\alpha(p)=
\begin{cases}
1, & \text{if } p^{e_\alpha(p)}\mid l_\alpha^{(p)},\
0, & \text{if } p^{e_\alpha(p)}\nmid l_\alpha^{(p)};
\end{cases}
\qquad (p\ne 2)
]
[
\tau_\alpha(2)=
\begin{cases}
2, & \text{if } \alpha=s(2),\ \sigma(\varphi_\alpha^{(2)})=1,\ t(2)=e_\alpha^{(2)}+1,\
2^{e_\alpha(2)}\mid l_s^{(2)},\ \text{and all}\
& \text{coefficients of the linear form }2^{-e_\alpha(2)}l_\alpha^{(2)}\text{ are odd};\
1, & \text{if }(s(2)+1-\alpha)\sigma(\varphi_\alpha^{(2)})(t(2)-e_\alpha(2))>1
\text{ and }2^{e_\alpha(2)+1}\mid l_\alpha^{(2)},\
& \text{if }\alpha=s(2)+1\text{ and }2^{e_\alpha(2)}\mid l_\alpha^{(2)};\
0, & \text{if }2^{e_\alpha(2)+1}\nmid l_\alpha^{(2)};\ \text{if }\alpha=s(2),\ \sigma(\varphi_\alpha^{(2)})=1,\ t(2)=e_\alpha(2)+1\
& \text{and }2^{e_\alpha(2)}\nmid l_\alpha^{(2)};\ \text{if }\alpha=s(2),\ \sigma(\varphi_\alpha^{(2)})=1,\ t(2)=e_\alpha(2)+1,\
& 2^{e_\alpha(2)}\mid l_\alpha^{(2)}\text{ and not all coefficients of the form }2^{-e_\alpha(2)}l_\alpha^{(2)}\text{ are odd};\
& \text{if }\alpha=s(2)+1\text{ and }2^{e_\alpha(2)}\nmid l_\alpha^{(2)};
\end{cases}
]
((p>q;\ \alpha=1,\ldots,s(p)+1)). Then
[
\begin{aligned}
S(f,l;q)=&
\frac{\displaystyle\prod_{p<q}\prod_{\alpha=1}^{s(p)+1}\tau_\alpha(p)^{\,n_\alpha(p)}}{2^{2n}}
\times S(f_1;4q)\times \
&\times \exp\left[-\frac{2\pi i}{4q}\sum_{p<q} Q(p)^{-1\ (\bmod\, p^{t(p)+1+(-1)^p})}\times Q(p)\times 2^{1+(-1)^p+1}\times\right.\
&\left.\times \left{2^{1+(-1)^p+1}\prod_{\alpha=1}^{s(p)} d_\alpha(p)\right}^{-1\ (\bmod\, p^{t(p)+1+(-1)^p})}
\times p^{\sum_{\alpha=1}^{s(p)} n_\alpha(p)e_\alpha(p)}
\times \bar\varphi^{(p)}(c_1,\ldots,c_n(p))\right],
\end{aligned}
\tag{6}
]
where (\bar\varphi^{(p)}) is a form algebraically reciprocal to (\varphi^{(p)}=\varphi^{(p)}(x_1,\ldots,x_n(p)));
(f_1=4f-\left(4-4^{2-\tau s(2)^{(2)}}\right)2^{e s(2)^{(2)}}\varphi_{s(2)}^{(2)}).
The sums (1) reduce to sums (2)
[
S_{g;b_1,\ldots,b_n}(f,l;q)=
\frac{1}{g^n}\exp\left[
2\pi i\,\frac{f(b_1,\ldots,b_n)+l(b_1,\ldots,b_n)}{q}
\right]S(g^2f,gL;q),
\tag{7}
]
where
[
\begin{aligned}
L=L(x_1,\ldots,x_n)
&=l(x_1,\ldots,x_n)+2f(b_1,\ldots,b_n;x_1,\ldots,x_n)\
&=\sum_{\alpha=1}^{n}\left(c_\alpha+2\sum_{\beta=1}^{n}a_{\alpha\beta}b_\beta\right)x_\alpha .
\end{aligned}
]
The formulas given above, together with estimates for Kloosterman sums ((^{4,6,7})), allow us to obtain the following useful proposition, which is a generalization and refinement of Kloosterman’s lemma ((^4)):
Theorem 4. Let (f=f(x_1,\ldots,x_n)) be an integral quadratic form with determinant (d); let (g>0,\ B_1,\ldots,B_n,\ C_1,\ldots,C_n,\ m,\ q>0,\ Q_1,\ Q_2) be integers; let
[
l_h=l_h(x_1,\ldots,x_n)=\sum_{\alpha=1}^{n}(hgB_\alpha+C_\alpha)x_\alpha .
]
Then
[
\left|
\sum_{Q_1\leq h^{-1}\,(\bmod\, q)\leq Q_2}^{\prime}
S(hg^2f,l_h;q)\exp\left[-2\pi i\,\frac{mh}{q}\right]
\right|
<
]
[
< \chi_\varepsilon q^{\frac{n+1}{2}+\varepsilon}
\left{\operatorname{g.c.d.}(2^{n+2}dg^2m-w,q)\right}^{\frac12}
\left{\operatorname{g.c.d.}(2^{3n}d^3g^{2n+4},q^{n+2})\right}^{\frac12};
\tag{8}
]
here the summation on the left is over all residues (h) of a reduced residue system ((\bmod\, q)) satisfying the condition (*) (Q_1\leq h^{-1}\,(\bmod\, q)\leq Q_2\,(\bmod\, q)); (w) is an integer depending only on (f,g) and (B_1,\ldots,B_n); (\varepsilon>0) is arbitrarily small; (\chi_\varepsilon>0) is a constant depending only on (n) and (\varepsilon); (w=0) if (B_1=\cdots=B_n=0).
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
6 IV 1960
REFERENCES
- K. F. Gauss, Tr. po teorii chisel, Moscow, 1959, p. 594.
- H. Weber, J. reine u. angew. Math., 74, 14 (1872).
- H. Minkowski, Gesammelte Abhandlungen, 1, Leipzig, 1911, S. 3.
- H. D. Kloosterman, Acta Math., 49, 407 (1926).
- B. W. Jones, The Arithmetic Theory of Quadratic Forms, 1950.
- H. Salié, Math. Zs., 34, 91 (1931).
- A. Weil, Proc. Nat. Acad. USA, 34, 204 (1948).
[
\text{* We say that } Q_1\leq z\leq Q_2\pmod q,\ \text{if there is } z_1\equiv z\pmod q
\text{ for which } Q_1\leq z_1\leq Q_2.
]