Download Elliptic curves do arise from ellipses

ELLIPTIC CURVES DO ARISE FROM ELLIPSES
MAHESH AGARWAL AND NARASIMHAMURTHI NATARAJAN
Abstract. We show that the locus of the foci of a family of ellipses
that are tangential to the sides of a triangle with one prescribed point of
tangency is an elliptic curve. We further show that every isomorphism
class of elliptic curves over an algebraically closed field K of char(K) 6=
2, 3 can be realized in this manner. On varying the triangle, one can get
all elliptic curves over an algebraically closed field K of char(K) 6= 2, 3.
1. Introduction
What are elliptic curves? For us, elliptic curves are curves of the form
y 2 = p(x) where p(x) is a cubic with no repeated roots. So are they related
to ellipses? Despite their names elliptic curves have not been directly linked
to ellipses. In fact (see [6], pg. 93) the general advice given to a student of
elliptic curves is: “Elliptic curves have (almost) nothing to do with ellipses,
so it is wise to put ellipses and conic sections out of your thoughts”. This
is perhaps due to the fact that even though “there is a connection between
ellipses and elliptic curves, but it’s not at all obvious and is the result of
a connected but distinctly nonlinear sequence of mathematical events” [1].
Historically, the journey from ellipses to elliptic curves follows a fascinating
though complex route via elliptic integrals (used to measure arc lengths of
ellipses) and elliptic functions (doubly periodic meromorphic functions). An
engaging account of these events is available in [1].
Why are elliptic curves important? Elliptic curves, show up in surprising
places. It plays a central role in Andrew Wiles proof of Fermat’s Last
Theorem (FLT): z n = xn + y n has no nontrivial zeros for n ≥ 3. In 1985,
Gerhard Frey, showed that if xn + y n = z n has a nontrivial solution for
n > 2 say (a, b, c) i.e. an + bn = cn then one can construct an elliptic curve
y 2 = x(x − an )(x + bn ) and this equation will have unusual properties. This
observation kicked off a sequence of discoveries finally leading to the proof
of FLT and opening up new avenues in number theory.
2010 Mathematics Subject Classification. Primary 30C10, 14H52.
Key words and phrases. Marden’s Theorem, elliptic curves, ellipses, duality.
The authors wish to thank Trevor Arnold, David James and Michael Lachance for
their feedback. They would also like to thank the anonymous member of the editorial
board whose suggestions deeply influenced the current form of the paper. They would
also like to acknowledge the use of GeoGebra and Maxima for some computations and
generating images.
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MAHESH AGARWAL AND NARASIMHAMURTHI NATARAJAN
Figure 1. The locus of the foci as P is fixed and Q is varied.
In this paper, we exhibit a simple and new way to realize elliptic curves
from ellipses. Given a triangle, there are circles that are tangential to all
three sides, namely, the incircle and the excircles. We can generalize this
idea to ellipses (conics). Using classical geometry or by an easy calculation
one can check that given two distinct points on any two sides of the triangle,
there is a unique conic tangential to the three sides of the triangle, touching
the triangle at the two prescribed points. Keeping one of the two points
fixed and moving the other point along the corresponding line, generates
a family of conics. We show that the loci of the foci of this family is an
elliptic curve. In addition we show that every elliptic curve defined over an
algebraically closed field K, char(K) 6= 2, 3 is obtained in this manner. The
route we take is Marden’s Theorem. This theorem provides a formula for
the foci as a function of the contact points.
2. Elliptic curves
We now give a formal definition of elliptic curves. Elliptic curves are
nonsingular cubic curves of the form
(1)
E : ax3 + bx2 y + cxy 2 + dy 3 + ex2 + f xy + gy 2 + hx + iy + j = 0
with at least one point on it (often referred to as a base point). Recall that
non-singular curves, are curves C(x, y) = 0 whose partial derivatives Cx and
Cy do not vanish simultaneously at any point on the curve. In most cases,
elliptic curves have a simpler representation. An elliptic curve E of the form
(1), is said to be defined over a field K if all its coefficients lie in K. Over any
field K with char(K) 6= 2, 3, any elliptic curve E using change of co-ordinates,
ELLIPTIC CURVES DO ARISE FROM ELLIPSES
3
can be written in a simpler form
E : y 2 = x3 + Ax + B
called the Weierstrass normal form. The actual change of variables can be
found in [5].
Taking a short detour, on a more conceptual level, elliptic curves are
smooth projective curves of genus one with a specified base point. Curves
of genus zero are conics and are well understood. For instance, if they have
one rational point (point defined over Q or field containing it) then they
have infinitely many rational points. Curves of genus greater than equal to
2, by a deep result of Faltings’ [2], have only finitely many rational points.
So elliptic curves are an interesting boundary case where they may have
finitely many or infinitely many rational points. We must also remark, that
above facts exhibit a beautiful interplay between a topological invariant of
the curve (its genus) and an arithmetic property (the number/existence of
rational points). The rational points on an elliptic curve come equipped with
a group structure. This group structure makes them particularly useful for
cryptography.
Coming back to our discussion, to each curve E, one can associate a
j-invariant (which is invariant under change of co-ordinates). In the Weierstrass normal form the j-invariant is given by
4A3
4A3 + 27B 2
The j-invariant is an important invariant of an elliptic curve and is closely
related to elliptic curve isomorphism. Two elliptic curves E (with variables
x, y) and E 0 (with variables x0 , y 0 ) are isomorphic over a field K if and only
if there is a one to one correspondence between points (x, y) and (x0 , y 0 )
defined only using operations within K. Such an isomorphism defines a
bijection between the set of rational points in E and E 0 . Since number
theorists are interested in rational points on curves, this property makes
isomorphism classes interesting to number theorists.
We now recall a result relating isomorphism classes and j-invariants:
j = 1728
Theorem 2.1 (Prop. 1.4, [7]). Two elliptic curves are isomorphic (over K)
if and only if they have the same j-invariant and for every j0 ∈ K, there is
an elliptic curve over K(j0 ) with j-invariant equal to j0 .
Here K denotes the algebraic closure of the field K and K(j0 ) is the field
obtained by adjoining j0 to K.
3. Marden’s Theorem
Marden’s theorem establishes a beautiful relationship between the relative
location of critical points of a polynomial vis-a-vis its roots. Let
f (z) = (z − z1 )m1 (z − z2 )m2 · · · (z − zk )mk
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MAHESH AGARWAL AND NARASIMHAMURTHI NATARAJAN
be a polynomial in a complex variable z. Since
f 0 (z) = f (z)[f 0 (z)/f (z)] = f (z)
d[log f (z)]
,
dz
the zeros of f 0 (z) fall into two categories - non-trivial if they are not zeros
of f (z) and trivial otherwise. Understanding the non-trivial zeros of f 0 (z)
amounts to studying the zeros of the rational function
k
F (z) :=
d[log f (z)] X mj
.
=
dz
z − zj
j=1
Motived by the above setup, Marden’s theorem concerns zeros of a rational
function in a more general setting, namely, F (z) = g(z)/h(z) whose deP
m
composition into partial fractions is of the form kj=1 z−zj j with mj being
arbitrary real constants. For this article we need only the case k = 3. We
now state Marden’s theorem:
Theorem 3.1 (Marden’s Theorem). The zeros z10 and z20 of the function
3
X
mj
F (z) =
z − zj
j=1
are the foci of the conic that touches the line segments (z1 , z2 ), (z2 , z3 ) and
(z3 , z1 ) in the points ζ3 , ζ1 and ζ2 that divide the segments in the ratios
m1 : m2 , m2 : m3 , m3 : m1 , respectively.
For example, in the special case when m1 = m2 = m3 = 1, it states that
the critical points of the polynomial p(z) = (z − z1 )(z − z2 )(z − z3 ) are the
foci of the ellipse that is inscribed in a triangle with vertices z1 , z2 , z3 and
tangential to the sides at the midpoints.
4. Elliptic curves from ellipses
We now show that the loci of the foci of conics tangential to three sides
of a triangle with one fixed point of contact is an elliptic curve with real
coefficients. For a given triangle, we show that every isomorphism class of
elliptic curves over K̄ (char(K) 6= 2, 3) can be obtained by this process when
the curve is formally interpreted with coefficients over K. Furthermore, we
show that every elliptic curve over K can be obtained by this process by a
proper choice of the triangle.
4.1. Let ABC be a triangle and P and Q be points on the line AB and
the line BC respectively. There a unique conic that touches the lines AB
and BC at P and Q respectively (see Figure 1). The following theorem
characterizes the loci of the foci of the conic as we vary the point Q along
BC while keeping P fixed.
ELLIPTIC CURVES DO ARISE FROM ELLIPSES
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Theorem 4.1. Let T be a triangle ABC. Let P be a fixed point on the line
AB and Q a point on BC. Keeping P fixed and varying Q generates a family
of conic sections that are tangential to side the AB at P and tangential to
the side BC at Q. The loci of the foci of these conic sections is an elliptic
curve for all but at most four choices of P .
Proof. Without loss of generality, by rotating and scaling the triangle, we
can assume A = −1, B = 1 and C = a + ib. Since P is on the line AB, it
is a real number. Let P = 2u − 1, u ∈ R. By Marden’s theorem the loci of
the foci of the conics described above, is the locus of z that satisfies
u
1−u
k
+
+
=0
z−A z−B z−C
as k varies over all real numbers. Note that as k varies, Q divides the
line BC in the ratio 1 − u : k. So varying k is same as varying Q. So
∃k ∈ R such that
u
1−u
k
+
+
= 0 where z3 = a + ib
z + 1 z − 1 z − z3
z−µ
k
⇔
+
= 0 where µ = 2u − 1
z 2 − 1 z − z3
(z − z3 )(z − µ)
⇔
+k =0
z2 − 1
(z − z3 )(z − µ)
⇔
is real
z2 − 1
⇔ (z − z3 )(z − µ)(z̄ 2 − 1) is real
Writing z = x + iy and setting the imaginary part equal to zero we get
0 = b x3 − 2 u x2 y − a x2 y + x2 y − 2 b u x2 + b x2
+b x y 2 + 4 a u x y − 2 a x y + 2 x y − b x − 2 u y 3 − a y 3 + y 3
(2)
+2 b u y 2 − b y 2 − 2 u y − a y + y + 2 b u − b
This is clearly a cubic. It remains to be shown it has at least one rational
point on it and it is non-singular. In the equation,
(z − z3 )(z − µ)
+k =0
z2 − 1
setting k = 0, we see that the points z3 and µ are on the curve and as
k → ∞, z 2 → 1. Hence, z = ±1 are on the curve. Hence the points A, B, C
and P lie on the curve. Not only do the vertices A and B lie on the curve, by
implicitly differentiating the above equation, we can verify that the tangent
at A and B go through C, i.e. the two sides are tangential at the vertices
A and B.
To find the points of singularity on the curve, note that z satisfies the
quadratic equation
(3)
(z − µ)(z − z3 ) + k(z 2 − 1) = 0
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MAHESH AGARWAL AND NARASIMHAMURTHI NATARAJAN
and the loci are the solutions parametrized by k. The loci would be singularity free except when the two roots coincide and this happens when the
discriminant:
z3 2 − 4 k µ z3 − 2 µ z3 + µ2 + 4 k 2 + 4 k = 0
Writing z3 = a + ib and separating the real and imaginary parts we get
µ2 − 4 a k µ − 2 a µ + 4 k 2 + 4 k − b2 + a2 = 0
−2 k µ − µ + a = 0
Eliminating k we get a fourth degree polynomial in µ. Hence for any given
triangle, there are at most four points P where the curve will be singular.
For all other choices of P , the resulting curve is nonsingular.
One can provide a strictly geometric characterization of the four points.
In any triangle, one can draw four circles that touch the three sides (one
incircle and three excircles). The point where these circles touch the base
AB are precisely the points where the ellipse becomes a circle, i.e. the two
foci coincide.
Remark 4.2. Though Marden’s theorem involves complex numbers and has
a geometric interpretation, the curve (2) defined above be formally viewed
as elliptic curve over any field K.
Remark 4.3. The elliptic curve in form (2) always has a 2-torsion point
since the tangents at A and B meet on the curve. Using the group structure
on the elliptic curve, 2A = 2B =⇒ 2(A − B) = 0.
While equation (2) shows that the curve is a cubic, it is not very illuminating. However, using the properties of this curve, we now give a simpler
representation of the elliptic curve obtained in Theorem 4.1.
Theorem 4.4. Suppose that P is distinct from A and B then the elliptic
curve in Theorem 4.1 after change of coordinates can be written in homogeneous co-ordinates as
(αX + βY )Z 2 + γXY Z + XY (X + Y ) = 0
where
b2 + a2 + 2 a + 1 (µ − 1)2
β = b2 + a2 − 2 a + 1 (µ + 1)2
γ = 4 (µ − a)
α =
Proof. To obtain the elliptic curve in the desired form, consider a projective
transformation that sends A to (1, 0, 0), B to (0, 1, 0), C (0, 0, 1) and P to (1,1,0). Since (1, 0, 0) is on the curve, there is no X 3 term in the homogeneous
co-ordinates. Similarly, there are no Y 3 and Z 3 terms. Since the sides
AC and BC of the triangle ABC touch the curve tangentially at A and
B respectively, there are no X 2 Z and Y 2 Z terms. Since, the point P =
ELLIPTIC CURVES DO ARISE FROM ELLIPSES
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(1, −1, 0) is on the curve the coefficient of X 2 Y is the same as that of XY 2 .
Normalizing by this coefficient we get curve
(4)
(αX + βY )Z 2 + γXY Z + XY (X + Y ) = 0
The formulas for α, β γ are a direct calculation, which we omit.
Remark 4.5. The coefficients α, β, γ can be given a geometric meaning
(α = AB 2 · P B 2 , etc.) which we do not pursue here.
On bringing equation (4) into Weierstrass normal form, one can check the
j-invariant is a non constant rational function. Hence we have the following
theorem:
Theorem 4.6. Every isomorphism class of elliptic curves over an algebraically closed field K can be realized as the loci of the foci of ellipses tangential to the three sides of a fixed triangle ABC for some common point of
tangency on the line AB.
Proof. Since j : P1 (K) → P1 (K) is a non constant rational function of u and
since K is algebraically closed, j it is surjective. Hence by Theorem 2.1, the
result follows.
In Theorem 4.6 the triangle ABC is fixed and by varying the point P
we are able to realize every isomorphism class of elliptic curves. If we allow
ourselves the freedom to vary the triangle ABC and the point P , we can
obtain every elliptic curve over an algebraically closed field as shown below:
Theorem 4.7. Every elliptic curve E defined over an algebraically closed
field K with char(K) 6= 2, 3 can be viewed as the loci of foci of conics tangential to some triangle ABC with a fixed point of contact P on the line
AB.
Proof. The transformation
X = −β X̂ − α Ẑ , Y = α X̂ − β Ẑ , Z = Ŷ
transforms (4) to
γ
Ŷ 2 Ẑ +
Ŷ X̂ − α Ẑ X̂ − β Ẑ = X̂ X̂ − α Ẑ X̂ − β Ẑ
α−β
For our purposes, it is sufficient to consider the case when γ = 0, i.e. a = µ.
In this case the curve
Ŷ 2 Ẑ = X̂ X̂ − α Ẑ X̂ − β Ẑ
is in Weierstrass form except for a shift in X̂-axis. We now give a constructive proof for obtaining the triangle ABC and point P . Starting from
the Weierstrass form for the curve E, shift the x axis so that the points on
y = 0 are at 0, α, and β. Since the roots are not repeated, α 6= 0, β 6= 0,
α 6= β. Now we can
solve the equations α = b2 + (µ + 1)2 (µ − 1)2 and
β = b2 + (µ − 1)2 (µ+1)2 for b and µ. This is possible since, β −α = 4b2 µ.
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MAHESH AGARWAL AND NARASIMHAMURTHI NATARAJAN
Solving for b2 and substituting in either of the equations, we get (fifth degree)
polynomial equation in µ. The solution for µ can then be used to compute
b. The locus of foci for triangle with vertices (a = µ, b), (−1, 0) and (1, 0)
with the point P = (µ, 0) generate an elliptic curve that is isomorphic to
the specified elliptic curve E.
Figure 2. Solid lines from ellipses, dashed from other conics
Remark 4.8. Not all points on an elliptic curve E are obtained as the loci
of the foci of ellipses as can be seen in Figure 2. To get the full elliptic curve
one has to consider all conics.
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[3] Dan Kalman, An elementary proof of Mardens theorem. Amer. Math. Monthly, 115
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3. American Mathematical Society, Providence, R.I., 1966.
[5] Joseph Silverman and John Tate, Rational points on elliptic curves. Undergraduate
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Sympos. Appl. Math., 62, 91-112, 2005
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#3, 443-551,1995
Department of Mathematics and Statistics, University of Michigan, Dearborn, Dearborn, MI 48128
E-mail address: [email protected]
Electrical and Computer Engineering Department, University of Michigan,
Dearborn, Dearborn, MI 48128
E-mail address: [email protected]