Csajbók, Zoltán Ernő

University of Debrecen Electronic Archive

Acta Mathematica Academiae Paedagogicae Ny´ıregyha´ziensis
23 (2007), 115–123
www.emis.de/journals
ISSN 1786-0091
ON THE MAPPINGS OF ELLIPTIC CURVES DEFINED
OVER Q INTO [0, 1)2
ZOLTA´N CSAJBO´K
Abstract. Let E be an elliptic curve defined over Q given by an affine
Weierstrass equation of the form
(1) E : y2 = x3 + ax+ b (a, b ∈ Z, x, y ∈ Q).
Reducing the elliptic curve (1) modulo a sufficiently large prime p, we
obtain an elliptic curve E˜p over Fp. Considering an infinite sequence of
elliptic curves E˜p, we map the point (x, y) of them into the unit square
[0, 1)2 via the mapping (x, y) 7→
(
x
p ,
y
p
)
.
We prove that the obtained cumulative point set contains a point se-
quence aligning a line when E/Q has an integral point, and point sequences
aligning lines of well defined number when E/Q has a rational point. In both
cases, these lines contain infinitely many points being strictly monotone in-
creasing or decreasing according to the L∞ norm, and these monotone point
sequences converge to well defined points.
1. Introduction
Let E be an elliptic curve defined over Q given in Weierstrass normal form
y2 = x3 + ax+ b (a, b ∈ Z, x, y ∈ Q, 4a3 6= 27b2).
If the prime p is sufficiently large, then its reduction modulo p
y2 = x3 + a˜x+ b˜ (a˜, b˜, x, y ∈ Fp)
is also an elliptic curve E˜p defined over the finite field Fp with p elements.
The finite abelian group E˜p(Fp) of Fp-rational points of E˜p has size
#E˜p(Fp) = p+ 1− ap
where |ap| < 2√p according to the Hasse-Weil theorem ([4], Chap. V, Theo-
rem 1.1).
2000 Mathematics Subject Classification. 14H52.
Key words and phrases. Elliptic curves, reduction of elliptic curves, normalization of
elliptic curves.
115
116 ZOLTA´N CSAJBO´K
The elliptic curve group E˜(Fp) can be studied at fixed p and varying E, or
conversely, at fixed E and varying p. If we fix p and vary E, then there are
only finitely many curves E over Fp up to equivalence, and Deuring’s theorems
contain detailed information of these curves [2].
It is much less known, however, the converse case, i.e., when E is fixed and
p varies.
Our approach to this problem is that we normalize the Fp-rational points
of E˜p simultaneously for all sufficiently large primes p, mapping them into the
unit square. In this paper we will study this normalized cumulative point set.
We prove that
• if E/Q has an integral point, then there is a line corresponding to that
point, which contains infinitely many points of the cumulative point
set (Theorem 6.1 (ii));
• if E/Q has a rational point, then there are lines of well defined number
corresponding to that point, and each of them also contains infinitely
many points of the cumulative point set (Theorem 7.1 (i) and (ii)).
Furthermore, in both cases, the point sequences lying on these lines are
strictly monotone increasing or decreasing according to the L∞ norm, and
they converge to well defined points (Theorem 6.1 (iii) and Theorem 7.1 (iii)).
2. Basic notation
Considering a fixed modulusm > 1, the finite residue ring Z/mZ is identified
with the set {0, 1, . . . ,m− 1}.
Note that if m = p is a prime, then Fp = Z/pZ is a field of p elements.
Throughout the paper, if u, v ∈ Z with gcd(v,m) = 1 then u/v (mod m)
denotes that (unique) w ∈ {0, 1, . . . ,m− 1} for which u ≡ vw (mod m) holds.
Let rp denote the natural reduction map of the p-integral elements Zvp of Q
onto Fp, i.e., rp(u/v) := u/v (mod p), where u, v ∈ Z with gcd(v, p) = 1. Note
that if a,m ∈ Z with m > 1, 0 < |a| < m, then a (mod m) = a if a > 0, and
a (mod m) = m+ a if a < 0.
For integers n ≥ 1 let ϕ(n) denote the Euler phi function, i.e.
ϕ(n) = #{x ∈ Z | 1 ≤ x ≤ n, gcd(x, n) = 1}.
For (x1, x2) ∈ R2 let ‖(x1, x2)‖ denote the L∞ norm of (x1, x2), i.e.
‖(x1, x2)‖ := max{|x1|, |x2|}.
3. Reduction of an elliptic curve over Q modulo p
Consider an elliptic curve E defined over Q given by an affine Weierstrass
equation of the form
(2) E/Q : y2 = x3 + ax+ b (a, b ∈ Z, x, y ∈ Q)
with discriminant ∆. Put
pmin := max{3, |a|, |b|, P (∆)},
ON THE MAPPINGS OF ELLIPTIC CURVES. . . 117
where P (∆) is the greatest prime divisor of ∆.
Reducing the elliptic curve (2) modulo a prime p > pmin, we obtain an
elliptic curve E˜p over Fp of the form
(3) E˜p : y
2 = x3 + a˜x+ b˜ (a˜, b˜, x, y ∈ Fp),
where a˜ = rp(a), b˜ = rp(b).
The elliptic curve E˜p is nonsingular for all primes p > pmin.
4. Reduction of the points of an elliptic curve over Q modulo p
An elliptic curve E(Q) can be written as a union of its affine part and the
point at infinity. A reduction modulo p, however, cannot map each point of
A2(Q) into A2(Fp). Namely, if P = (x, y) ∈ A2(Q), then its reduction modulo
p is in A2(Fp) if and only if the rational numbers x and y are p-integral.
For the rational points of the curve (2) we have the following statement.
Proposition 4.1 ([5], pp. 68-69). Let (x, y) be a Q-rational point on the curve
E/Q of the form (2). Then
x =
x0
e2
, y =
y0
e3
(4)
for some integers x0, y0, e with e > 0 and gcd(x0, e) = gcd(y0, e) = 1.
Hence, if P is a rational point, then either both coordinates are p-integral
(vp(x), vp(y) ≥ 0), or p divides the denominators of both x and y (vp(x), vp(y) <
0). Thus we can construct the reduction homomorphism as follows:
(5) Raffp : E(Q)→ E˜p(Fp)
P = (x, y) 7→

(rp(x), rp(y)), if P ∈ E(Q) \ {O}, vp(x), vp(y) ≥ 0;
O, if P = O, or P ∈ E(Q) \ {O},
vp(x), vp(y) < 0.
where E/Q and P = (x, y) are in the form (1) and (4) respectively.
5. Mappings of the points of E(Q) and E˜p(Fp) into [0, 1)2
Consider an infinite sequence of elliptic curves E˜p of the form (3) for primes
p > pmin, and define the following sets:
Êp :=
{(
x
p
,
y
p
)
| (x, y) ∈ E˜p(Fp) \ {O}, p > pmin
}
,
Ê :=
⋃
p>pmin
Êp,
and the natural mappings
Ωp : E˜p(Fp) \ {O} → Êp ⊂ Ê ⊂ [0, 1)2, (x, y) 7→
(
x
p
,
y
p
)
.
118 ZOLTA´N CSAJBO´K
Some basic elementary properties of the mappings Ωp, and the sets Êp, Ê
are summarized in the following proposition.
Proposition 5.1. (i) The points of Êp are symmetric to the line y =
1
2
for all primes p > pmin.
(ii) If x0y0 6= 0, p1 6= p2 are different primes, p1, p2 > pmin, and P =
(x0, y0) ∈ E(Fp1) ∩ E(Fp2), then
Ωp1(x0, y0) 6= Ωp2(x0, y0).
Moreover, the points Ωp1(x0, y0),Ωp2(x0, y0) are not on a line parallel
either to the x-axis, or to the y-axis.
(iii) If x0y0 6= 0, then for each P ∈ Ê there is exactly one prime p that
P ∈ Êp.
(iv) If x0y0 6= 0 then the cardinality of the preimage of P ∈ Ê is exactly
one, i.e, #Ω−1p (P ) = 1.
(v) If a rational point Q =
(
x0
p
, y0
p
)
∈ [0, 1)2 with x0y0 6= 0 does not belong
to the image of the mapping Ωp, i.e., Q /∈ Êp for the prime p, then
Q /∈ Ê.
Proof. (i) It follows from the fact that (x, y) ∈ E implies (x,−y) ∈ E.
(ii) If P = (x0, y0) ∈ E(Fp1) ∩ E(Fp2), then x, y <min{p1, p2}, thus
Ωp1(x0, y0) =
(
x0
p1
,
y0
p1
)
6=
(
x0
p2
,
y0
p2
)
= Ωp2(x0, y0),
since p1 6= p2, especially x0p1 6= x0p2 and
y0
p1
6= y0
p2
.
(iii) If p1, p2 are primes, then there are not exist integers 0 < x0 < p1 and
0 < x′0 < p2 such that
x0
p1
=
x′0
p2
, because x0 · p2 = x′0 · p1 implies, e.g., p1 | x0
which is, however, contradicts the assumption that 0 < x0 < p1.
(iv) It follows from part (iii).
(v) If Q =
(
x0
p
, y0
p
)
/∈ Êp for the prime p, then, by definition, Q /∈ Êp′ for
any other primes p′ as well. ¤
Let E(Q)vp denote the subset of E(Q) which consists of all points of E(Q)
whose both coordinates are p-integral, that is
E(Q)vp := {(x, y) | (x, y) ∈ E(Q) \ {O}, vp(x), vp(y) ≥ 0, p > pmin}.
Restricting the reduction map Raffp to the subset E(Q)vp
Raffp |E(Q)vp : E(Q)vp → E˜p(Fp), (x, y) 7→ (rp(x), rp(y)),
we get the map
Rp := Ωp ◦Raffp : E(Q)vp → Ê, (x, y) 7→
(
rp(x)
p
,
rp(y)
p
)
,
with Rp(E(Q)vp) = Ωp(Raffp (E(Q)vp)) ⊂ Êp ⊂ Ê ⊂ [0, 1)2.
ON THE MAPPINGS OF ELLIPTIC CURVES. . . 119
6. On the structure of Ê when E(Q) has an integral point
If E(Q) has an integral point, then Ê has the following properties. Note
that if (x0, y0) is an integral point of E(Q), then
Raffp (x0, y0) = (x0(mod p), y0(mod p)) ∈ E˜p(Fp)
for all primes p > pmin.
Theorem 6.1. Let (x0, y0) be a fixed integral point of E(Q), where x0, y0 ∈ Z,
and x0y0 6= 0.
(i) The points Rp(x0, y0) =
(
x0 (mod p)
p
, y0 (mod p)
p
)
are different for all primes
p > max{|x0|, |y0|, pmin}.
(ii) The points Rp(x0, y0) =
(
x0 (mod p)
p
, y0 (mod p)
p
)
lie on the lines
y = y0
x0
x, if x0 > 0, y0 > 0,
y − 1 = y0
x0
x, if x0 > 0, y0 < 0,
y − 1 = y0
x0
(x− 1), if x0 < 0, y0 < 0,
y = y0
x0
(x− 1), if x0 < 0, y0 > 0,
for primes p > max{|x0|, |y0|, pmin}.
(iii) The infinite sequence of points
{Rp(x0, y0)}p>max{|x0|,|y0|,pmin}
is convergent, and Rp(x0, y0)→ (x∗, y∗) with
(x∗, y∗) =

(0, 0), if x0 > 0, y0 > 0;
(0, 1), if x0 > 0, y0 < 0;
(1, 0), if x0 < 0, y0 > 0;
(1, 1), if x0 < 0, y0 < 0.
Furthermore, in each case
‖(x∗, y∗)−Rp2(x0, y0)‖ < ‖(x∗, y∗)−Rp1(x0, y0)‖
for p2 > p1.
Proof. Statements (i), (ii), (iii) will be proved only for the case when
x0 > 0, y0 < 0. For other cases, the proof can be carried out similarly.
(i) For x0 > 0, y0 < 0, we have
Rp1(x0, y0) =
(
x0
p1
, 1 +
y0
p1
)
6=
(
x0
p2
, 1 +
y0
p2
)
= Rp2(x0, y0).
(ii) For x0 > 0, y0 < 0, the point Rp(x0, y0) =
(
x0
p
, 1 + y0
p
)
is on the line
y = y0
x0
x+ 1, because
1 +
y0
p
=
y0
x0
x0
p
+ 1.
120 ZOLTA´N CSAJBO´K
(iii) For x0 > 0, y0 < 0 and p > max{x0, |y0|, pmin}, we have
x0 (mod p)
p
=
x0
p
→ 0 (p→∞),
y0 (mod p)
p
= 1 +
y0
p
→ 1 (p→∞).
Furthermore, if p2 > p1 > max{x0, |y0|, pmin}, then
‖(0, 1)−Rp2(x0, y0)‖ < ‖(0, 1)−Rp1(x0, y0)‖
if and only if ∥∥∥∥(x0 (mod p2)p2 , 1− y0 (mod p2)p2
)∥∥∥∥
≤
∥∥∥∥(x0 (mod p1)p1 , 1− y0 (mod p1)p1
)∥∥∥∥ .
However,
max
{
x0 (mod p2)
p2
, 1− y0 (mod p2)
p2
}
= max
{
x0
p2
, 1− p2 + y0
p2
}
< max
{
x0
p1
, 1− p1 + y0
p1
}
= max
{
x0 (mod p1)
p1
, 1− y0 (mod p1)
p1
}
.
¤
7. On the structure of Ê when E(Q) has a rational point
Throughout this section, let (x0
e2
, y0
e3
) be a fixed rational point of E(Q), where
x0, y0, e ∈ Z with e > 1, gcd(x0, e) = gcd(y0, e) = 1, x0y0 6= 0.
If E(Q) has a rational point, then Ê has the following properties.
Theorem 7.1. (i) For all primes p > max{|x0|, |y0|, e3, pmin} the points
Rp
(
x0
e2
, y0
e3
)
lie on lines of the form
li : y − Ai = y0
ex0
(x−Bi) ,(6)
where (Ai, Bi) = (
i
e3
, ix0/y0 (mod e
2)
e2
) with 0 < i < e3, gcd(i, e3) = 1.
(ii) The number of li is ϕ(e
3), and each of them contains infinitely many
points of the form Rp(
x0
e2
, y0
e3
).
(iii) For all i the infinite sequence of the points{
Rp
(x0
e2
,
y0
e3
)}
p>max{|x0|,|y0|,e3,pmin}
on the line li is convergent, and
Rp
(x0
e2
,
y0
e3
)
→ (Bi, Ai) (p→∞).(7)
ON THE MAPPINGS OF ELLIPTIC CURVES. . . 121
Furthermore, for primes p2 > p1 > max{|x0|, |y0|, e3, pmin} with
p1 ≡ p2 ≡ −y0/i (mod e3)
we have ∥∥∥(Bi, Ai)−Rp2 (x0e2 , y0e3)∥∥∥ < ∥∥∥(Bi, Ai)−Rp1 (x0e2 , y0e3)∥∥∥ ,
and the sequences
{
rp(x0/e2)
p
}
and
{
rp(y0/e3)
p
}
are strictly monotone increasing
or decreasing if x0 < 0 or x0 > 0 and y0 < 0 or y0 > 0 respectively.
To prove the theorem, we need the following statement.
Lemma 7.2. With the notation of Theorem 7.1, for p > max{|x0|, |y0|} with
i = −y0/p (mod e3) we have
y0
e3
(mod p) = pAi +
y0
e3
,
x0
e2
(mod p) = pBi +
x0
e2
.
Proof. We are going to proof only the equation x0
e2
(mod p) = pBi +
x0
e2
, the
other one can be proved similarly.
Since i = −y0/p (mod e3), the congruence i ≡ −y0/p (mod e2) holds, thus
Bi =
−x0/p (mod e2)
e2
. It only remains to show that
(x0/e
2 (mod p)) · e2 ≡ (−x0/p (mod e2)) · p+ x0 (mod p), and
0 ≤ (−x0/p (mod e2)) · p+ x0 < pe2.
The congruence obviously holds, and the inequalities follow from the facts
that 1 ≤ −x0/p (mod e2) ≤ e2 − 1 and −p < x0 < p.
¤
Proof of the theorem. We return to the proof of Theorem 7.1.
(i) Clearly, Rp
(
x0
e2
, y0
e3
) ∈ Êp ⊂ Ê.
Substituting the number
y0
e3
(mod p)
p
for y in the left side of equation (6), and
applying Lemma 7.2 we have:
y0
e3
(mod p)
p
− Ai =
pAi +
y0
e3
p
− Ai = y0
pe3
.
Substituting the number
x0
e2
(mod p)
p
for x in the right side of equation (6),
and applying Lemma 7.2 we have:
y0
ex0
( x0
e2
(mod p)
p
−Bi
)
=
y0
ex0
(
pBi +
x0
e2
p
−Bi
)
=
y0
pe3
.
(ii) It is obvious that #{li | 0 < i < e3, gcd(i, e3) = 1} = ϕ(e3).
Furthermore, there are exist infinitely many primes for which p ≡
−y0/i (mod e3) holds because of the Dirichlet’s theorem on primes in arith-
metical progressions ([1], Chap. 7).
122 ZOLTA´N CSAJBO´K
(iii) It follows from the definition of Rp and Lemma 7.2, e.g.,
for p > max{|x0|, |y0|} with i = −y0/p (mod e3) we have
rp(
x0
e2
)
p
=
x0
e2
(mod p)
p
=
pBi +
x0
e2
p
= Bi +
x0
pe2
→ Bi (p→∞).(8)
From (8) it follows immediately that the sequence
{
rp(x0/e2)
p
}
strictly mono-
tone increase or decrease depending on x0 < 0 or x0 > 0.
By Lemma 7.2 we get∥∥∥∥(Bi − x0e2 (mod p2)p2 , Ai −
y0
e3
(mod p2)
p2
)∥∥∥∥
=
∥∥∥∥(− x0p2e2 ,− y0p2e3
)∥∥∥∥ = max{∣∣∣∣− x0p2e2
∣∣∣∣ , ∣∣∣∣− y0p2e3
∣∣∣∣}
< max
{∣∣∣∣− x0p1e2
∣∣∣∣ , ∣∣∣∣− y0p1e3
∣∣∣∣} = ∥∥∥∥(− x0p1e2 ,− y0p1e3
)∥∥∥∥
=
∥∥∥∥(Bi − x0e2 (mod p1)p1 , Ai −
y0
e3
(mod p1)
p1
)∥∥∥∥ .
¤
Remark 7.3. Theorem 7.1 is also true for primes p < e3 with gcd(p, e) = 1.
However, we are interested in the asymptotic behavior of the sequence
Rp
(
x0
e2
, y0
e3
)
. Thus, for the sake of simplicity of the treatment, we can put
aside finitely many primes.
Acknowledgements. I would like to thank Prof. Dr. A. Petho˝ for
the basic idea of this article and helpful discussions and suggestions, and
Dr. T. Herendi, Dr. T. Miha´lydea´k for effective help.
I also would like to thank the anonymous referee for careful reading of the
manuscript and for many useful comments and suggestions.
References
[1] T. M. Apostol. Introduction to Analytic Number Theory. Undergraduate Texts in Math-
ematics. New York-Heidelberg-Berlin: Springer-Verlag, 1976.
[2] M. Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenko¨rper. Abh.
Math. Semin. Hansische Univ., 14:197–272, 1941.
[3] A. W. Knapp. Elliptic Curves.Mathematical Notes (Princeton). 40. Princeton University
Press, 1992.
[4] J. H. Silverman. The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106.
Springer-Verlag, 1986.
[5] J. H. Silverman and J. Tate. Rational Points on Elliptic Curves. Undergraduate Texts
in Mathematics. Springer-Verlag, 1992.
Received January 17, 2007.
ON THE MAPPINGS OF ELLIPTIC CURVES. . . 123
Department of Health Informatics,
Health College Faculty,
University of Debrecen
So´sto´i u. 2–4.,
H-4400 Ny´ıregyha´za,
Hungary
E-mail address: csajzo@freemail.hu; csajzo@de-efk.hu


On the mappings of elliptic curves defined over Q into [0,1)²

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