Abstract
Full Text
UDC 511.26
MATHEMATICS
S. A. STEPANOV
CONGRUENCES MODULO A POWER OF A PRIME NUMBER
(Presented by Academician I. M. Vinogradov on 1 X 1968)
1. Let
[
F(x,y)=y^n+P_1(x)y^{n-1}+\cdots+P_{n-1}(x)y+P_n(x)
]
be an absolutely irreducible polynomial with integer rational coefficients, and let (p) be a prime number. In the present paper we study the number of solutions of the congruence
[
F(x,y)\equiv 0 \pmod {p^m},
\tag{1}
]
where (x) and (y) run through an incomplete system of residues.
In the special case, namely for the congruence (x^2+y^2\equiv 1\pmod {p^m}), such an investigation was carried out by L. P. Postnikova ((^1)). This investigation was based essentially on the fact that the congruence has the form (y^n\equiv f(x)\pmod {p^m}). The congruence (ax^3+y^3\equiv 1\pmod {p^m}), considered by P. D. Varbanets ((^2)), has the same form. We shall show in what way an investigation of the general case (1) can be carried out.
In what follows, by (D(x)) we shall denote the discriminant of the polynomial (F(x,y)), and by (c_1,c_2,\ldots) positive constants depending only on the polynomial (F(x,y)). Denote by (N(F,p)) the number of solutions of the congruence (F(x,y)\equiv 0\pmod p). A well-known result of A. Weil ((^3)) asserts that for the quantity (N(F,p)) the estimate
[
|N(F,p)-p|<c_1\sqrt p.
]
Theorem. Let (m>c_2) be a natural number,
[
p^{(m-1)/{[c_3(m-1)-n]-1}}\leq T_1\leq p^m,\qquad 1\leq T_2\leq p^m.
]
Denote by (A(T_1,T_2)) the number of solutions of the congruence (1) such that (D(x)\not\equiv 0\pmod p), for which (0\leq x\leq T_1-1), (0\leq y\leq T_2). Then for the quantity (A(T_1,T_2)) we have the expression
[
A(T_1,T_2)=\frac{T_1T_2}{p^m}\,\frac{N(F,p)+O(1)}{p}
+O\left(e^{7m\ln^2 m}T_1^{-1/12m^3\ln 12m^3}\right),
]
where the constants entering the symbol (O) depend only on the polynomial (F).
2. Suppose that ((x_0,y_0)) is a solution of the congruence (F(x,y)\equiv 0\pmod p) such that
[
D(x_0)\not\equiv 0\pmod p.
\tag{2}
]
Lemma 1. Let condition (2) be fulfilled. Then for any rational integer (t), (0\leq t\leq p^{m-1}-1), there exists a unique solution of the congruence
[
F(x_0+pt,y)\equiv 0\pmod {p^m}
]
such that (y\equiv y_0\pmod p).
This lemma is a variant of the well-known Hensel lemma (see ((^4)), p. 365).
Lemma 2. Let (\theta) be a primitive integer of the field of algebraic numbers (K), let (f(y)) be its minimal polynomial, and let (p) be a prime number not dividing the discriminant of the polynomial (f). Suppose that modulo (p) there is the factorization
[
f(y)\equiv f_1(y)\cdots f_s(y)\pmod p,
]
where (f_1,\ldots,f_s) are irreducible integral polynomials of degrees (n_1,\ldots,n_s), respectively, pairwise distinct modulo (p). Then the decomposition of the number (p) into a product of prime divisors of the field (K) has the form
[
p=\mathfrak p_1\cdots \mathfrak p_s,
]
where the distinct prime divisors (\mathfrak p_1,\ldots,\mathfrak p_s) have degrees (n_1,\ldots,n_s), respectively, and moreover (f_i(\theta)\equiv 0\pmod{\mathfrak p_i}) for each (i=1,2,\ldots,s).
For the proof see [4], p. 271, Theorem 8.
Lemma 3. There exists a polynomial of degree not exceeding (m-1)
[
y(t)=\sum_{\nu=0}^{m-1} e_\nu(x_0,y_0)p^{\lambda_\nu}t^\nu,
]
for which (e_\nu(x_0,y_0)), (\nu=0,1,\ldots,m-1), are integral rational numbers relatively prime to (p), and (\lambda_\nu) are nonnegative integral rational numbers satisfying the inequalities (\lambda_\nu\ge \nu), (\nu=0,1,\ldots,m-1), such that
[
F(x_0+pt,y(t))\equiv 0\pmod{p^m}
]
and (y(t)\equiv y_0\pmod p).
Proof. We have
[
F(x_0+pt,y)=y^n+P_1(x_0+pt)y^{n-1}+\cdots+P_{n-1}(x_0+pt)y+
+P_n(x_0+pt).
]
Put (z=pt) and consider (F(x_0+z,y)) as a polynomial in (z,y). Denote by (f(y)) an irreducible divisor of the polynomial (F(x_0,y)) for which (y_0) is a root of the congruence (f(y)\equiv 0\pmod p). Let (\theta) be a root of the polynomial (f(y)) in the finite extension (K=R(\theta)) of the field of rational numbers (R). Denote by (a_0) a root of the congruence (f(y)\equiv 0\pmod{p^m}). By Lemma 2, in view of condition (2), we obtain that
[
\theta\equiv a_0\pmod{\mathfrak p^m}, \tag{3}
]
where (\mathfrak p) is a prime divisor of the field (K) of degree one occurring in (p).
Consider the equation (F(x_0+z,y)=0) over the ring of integral rational numbers. Let (y=v(z)) be a solution of this equation in a neighborhood of (z=0), and let (v(0)=\theta). By condition (2), the point (z=0) will be regular, and (v(z)) in some neighborhood of this point expands into a power series
[
v(z)=\theta+\sum_{\nu=1}^{\infty}\theta_\nu z^\nu .
]
It is easy to show that all (\theta_\nu\in K). Further, from the Eisenstein theorem ([5], p. 155), again using condition (2), it is easy to obtain that the coefficients (\theta_\nu) of the expansion of (v(z)) into a power series will be (\mathfrak p)-integral. Hence, in view of (3), we obtain that (v(pt)\equiv \sum_{\nu=0}^{m-1} a_\nu t^\nu\pmod{\mathfrak p^m}), where (a_\nu) are integral rational numbers, and that (F(x_0+pt,v(pt))\equiv 0\pmod{p^m}). Representing now (a_\nu), (\nu=0,1,\ldots,m-1), in the form (a_\nu=e_\nu(x_0,y_0)p^{\lambda_\nu}) and denoting (v(pt)=y(t)), we obtain the assertion of the lemma.
The following lemma belongs to Dudzi.
Lemma 4. Let the function
[
a(z)=\sum_{\nu=0}^{\infty}a_\nu z^\nu
]
be regular in the unit disk (|z|<1), and let all singularities of this function for (|z|=1) be algebraic. Let (n) be the number of these singularities.
Then, for (\nu\ge \nu_0),
[
0<Ar^\nu\le |a_\nu|+|a_{\nu+1}|+\cdots+|a_{\nu+n-1}|\le Br^\nu,
]
where (r) is a rational number distinct from a negative integer, and (A) and (B) are certain constants independent of (\nu).
From this lemma it follows directly that if the series
[
v(z)=\theta+\sum_{\nu=1}^{\infty}\theta_{\nu}z^{\nu}
]
has radius of convergence (\lambda>0), then for (\nu\geq \nu_0) the estimate (|\theta_\nu|\leq c_4\nu^{c_5}\lambda^{-\nu}) holds, and among (n) consecutive coefficients (\theta_\nu) at least one is nonzero. The same is also true for all conjugate numbers (\theta_\nu) in the field (K). Then it is easy to obtain that, for (m>c_6(\nu_0+n+2)+1=c_2), (\nu_0\leq \nu\leq m/c_6), for (a_\nu\ne0) the estimate (v_p(a_\nu)\leq c_7\nu) holds, where (v_p(\alpha)) is the (p)-adic exponent of the number (\alpha). Thus we have proved:
Lemma 5. Let the polynomial
[
y(t)=\sum_{\nu=0}^{m-1} e_\nu(x_0,y_0)p^{\lambda_\nu}t^\nu
]
be defined by Lemma 3, and let (m>c_2). Then for (\nu_0\leq \nu<m/c_6-n), and for any (x_0,y_0) that are solutions of the congruence (F(x,y)\equiv0\pmod p) and satisfy condition (2), the estimate
[
\min(\lambda_\nu,\lambda_{\nu+1},\ldots,\lambda_{\nu+n-1})\leq c_6(\nu+n)\qquad (c_6\geq c_7)
]
holds.
- We proceed to the proof of the theorem. Suppose that (T_1=pU-1). On the basis of Lemma 3 it is clear that (A(pU-1,T_2)) is equal to the number of fractional parts
[
\left{\bigl[e_0(x_0,y_0)+e_1(x_0,y_0)p^{\lambda_1}t+\cdots+e_{m-1}(x_0,y_0)p^{\lambda_{m-1}}t^{m-1}\bigr]/p^m\right},
]
where (x_0,y_0) run through all solutions of the congruence (F(x,y)\equiv0\pmod p) such that (D(x_0)\not\equiv0\pmod p), and (t=0,1,\ldots,U-1), which fall in the interval ((0,T_2/p^m)).
Denote by (\chi(t)) the characteristic function of the interval ((0,T_2/p^m)). Then
[
A(pU-1,T_2)=
\sum_{\substack{x_0\ y_0\ D(x_0)\not\equiv0\ (\mathrm{mod}\ p)}}
\chi\left(\frac{e_0+e_1p^{\lambda_1}t+\cdots+e_{m-1}p^{\lambda_{m-1}}t^{m-1}}{p^m}\right).
]
By a known method, the study of the number of fractional parts is reduced to estimating the modulus of the trigonometric sum
[
\sum_{t=0}^{U-1}\exp\left(2\pi i n\frac{e_0+e_1p^{\lambda_1}t+\cdots+e_{m-1}p^{\lambda_{m-1}}t^{m-1}}{p^m}\right).
]
Put (\mu=[(m-1)/c_6-n]). From the condition (m>c_6(\nu_0+n+2)+1) it follows that (\mu\geq\nu_0+2). Moreover, it is obvious that (\mu\leq(m-1)/c_6-n). Then by Lemma 5
[
\min(\lambda_\mu,\lambda_{\mu+1},\ldots,\lambda_{\mu+n-1})\leq c_6(\mu+n)\leq m-1.
]
Applying to the trigonometric sum (4) the estimate of I. M. Vinogradov (see (7), p. 389), which we shall carry out with respect to the coefficient of (t^\nu), where (\nu) is equal to that one of the numbers (\mu,\mu+1,\ldots,\mu+n-1) for which (\lambda_\mu\leq m-1), gives
[
\left|
\sum_{t=0}^{U-1}
\exp\left{2\pi i n\frac{e_0+e_1p^{\lambda_1}t+\cdots+e_{m-1}p^{\lambda_{m-1}}t^{m-1}}{p^m}\right}
\right|
\leq
]
[
\leq e^{7m\ln^2 m}n^{27/32m^2\ln 8m^2}U^{1-1/3m^3\ln 12m^3}.
]
Then we obtain
[
A(pU-1,T_2)=\frac{T_2U}{p^m}
\sum_{\substack{x_0\ y_0\ D(x_0)\not\equiv0\ \mathrm{mod}\ p}}
+O!\left(e^{7m\ln^2 m}pU^{1-1/6m^3\ln 12m^3}\right).
]
Representing (T_1) in the form (T_1=pU-r), where (1\le r\le p-1), it is then easy to show that
[
A(T_1,T_2)=\frac{T_1T_2}{p^m}\frac{N(F,p)}{p}+O(1)+O\left(e^{7m\ln^2 m}T_1^{-1/12m^3\ln 12m^3}\right).
]
The theorem is thereby proved.
Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
1 X 1968
CITED LITERATURE
(^{1}) L. P. Postnikova, Matem. sborn., 65, no. 2, 228 (1964).
(^{2}) P. D. Varbanets, Analytic Theory of Congruences Modulo a Power of a Prime Number, Abstract of Candidate’s Dissertation, Saratov, 1967.
(^{3}) A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Act. Sci. Ind., 1041, Paris, 1948.
(^{4}) Z. I. Borevich, I. R. Shafarevich, Number Theory, Moscow, 1964.
(^{5}) G. Pólya, G. Szegő, Problems and Theorems in Analysis, 2, Moscow, 1956.
(^{6}) M. Tsuji, Japan. J. Math., 3, No. 1–2, Tokyo, 69 (1926).
(^{7}) I. M. Vinogradov, Selected Works, Publishing House of the Academy of Sciences of the USSR, 1952.