Elliptic Curves
Dr. Carmen Bruni
University of Waterloo
November 4th, 2015
Dr. Carmen Bruni Elliptic Curves
Revisit the Congruent Number Problem
Congruent Number Problem
Determine which positive integers N can be expressed as the areaof a right angled triangle with side lengths all rational.
For example 6 is a congruent number since it is the area of the3− 4− 5 right triangle.
Dr. Carmen Bruni Elliptic Curves
From Triangles to Curves
Now, we’re going to take the information about our triangleand get a new equation which will turn out to represent acurve in the real plane.
Let x2 + y2 = z2 and xy = 2N for rationals x , y , z and somecongruent number N.
Adding and subtracting 2xy = 4N to the first equation gives
x2 + 2xy + y2 = z2 + 4N x2 − 2xy + y2 = z2 − 4N
Factoring and dividing by 4 gives the two equations(x + y
2
)2
= (z/2)2 + N
(x − y
2
)2
= (z/2)2 − N
Dr. Carmen Bruni Elliptic Curves
From Triangles to Curves
With the equations(x + y
2
)2
= (z/2)2 + N
(x − y
2
)2
= (z/2)2 − N
we multiply these two equations together gives(x + y
2
)2
·(x − y
2
)2
= ((z/2)2 + N)((z/2)2 − N)((x + y
2
)·(x − y
2
))2
= (z/2)4 + N(z/2)2 − N(z/2)2 − N2
((x − y)(x + y)
4
)2
= (z/2)4 − N2
(x2 − y2
4
)2
= (z/2)4 − N2
Dr. Carmen Bruni Elliptic Curves
From Triangles to Curves
Letting u = z/2 and v = (x2 − y2)/4, the previous equationbecomes
v2 = u4 − N2
Multiplying by u2 gives
(uv)2 = (u2)3 − N2u2
Finally, we let y = uv and x = u2 which gives us the equation
y2 = x3 − N2x
We call such curves where y2 equals a cubic in x an EllipticCurve (provided the discriminant is nonzero; this is the casefor cubics associated to the Congruent Number Problem).
Dr. Carmen Bruni Elliptic Curves
Examples of an Elliptic Curve
Let’s look at examples of elliptic curves. What do they looklike on the real plane?
Let’s try to draw y2 = x3 − x first by drawing y = x3 − x andthen trying to draw the elliptic curve.
Dr. Carmen Bruni Elliptic Curves
Drawing y = x3 − x
First, note that y = x3 − x = x(x − 1)(x + 1) and so theequation has three zeroes at x = 0,±1.
Now let’s break this curve into four intervals and see whathappens in each interval y = x3 − x = x(x − 1)(x + 1).
Between −∞ and −1, the function is negative.
Between −1 and 0, the function is positive.
Between 0 and 1, the function is negative.
Between 1 and ∞, the function is positive.
Lastly, the curve should look smooth with no breaks.
Dr. Carmen Bruni Elliptic Curves
The Cubic Curve y = x3 − x
Here is the picture (Using Desmos.com)
Dr. Carmen Bruni Elliptic Curves
The Elliptic Curve y 2 = x3 − x
What changes when we make the left hand side y2 instead ofy?
For almost all values of x , we will get not 1 but 2 outputvalues (the exceptions are the roots).
This means that we no longer have a function, rather a curve.
The cubic must be positive to have a real root! So all theareas where the picture is negative are gone.
The curve still has no breaks and is symmetric about thex-axis, that is, if I reflect the top half of the picture, it shouldmatch the bottom half.
The function should still be smooth (even at 1).
Dr. Carmen Bruni Elliptic Curves
The Cubic Curve y = x3 − x
Here is the picture (All graphs courtesy of Desmos.com)
Notice that the curve has two connected components!
Dr. Carmen Bruni Elliptic Curves
Connected Components
Note: In general, not all elliptic curves have two components.Some have one like y2 = x3 − 1:
However, the elliptic curves associated to the Congruent NumberProblem always have two connect components.
Dr. Carmen Bruni Elliptic Curves
Points on an elliptic curve
Elliptic curves have infinitely many real points.
As an example, y2 = x3 − x has infinitely many real points bynoticing that the cubic on the right is always positive whenx > 1 and hence we can find a y value by taking the squareroot.
So if we take x = 2, then we see that y2 = 23 − 2 = 6 and sothe point P = (2,
√6) and Q = (2,−
√6) are on the curve.
Dr. Carmen Bruni Elliptic Curves
Points on an elliptic curve
From the perspective of Diophantine equations, it isinteresting to ask: How many integer points are on ellipticcurves?
For the example y2 = x3 − x , it turns out that (±1, 0) and(0, 0) are the only integer points, though this is hardlyobvious.
How many rational points are on elliptic curves?
Above, the only rational points are also the integral ones.
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
With an elliptic curve, wecan actually describe a wayto, given two rational pointsP and Q, create a thirdrational point R.
Let’s begin with the ellipticcurve y2 = x3 − x + 1 forillustrative purposes.
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
Let’s take the pointsP = (−1.324, 0) andQ = (0, 1) (correct to threedecimal places).
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
Draw the line between Pand Q. It intersects thecurve in a third point asshown in the picture atcoordinates (1.895, 2.43).
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
Draw the vertical linethrough the point whichmust intersect the curve in athird point, in our case,R = (1.895,−2.43) (this isthe same as reflecting aboutthe x-axis).
Define P + Q = R for pointson an elliptic curve (notethat this isn’t just addingthe coordinates!)
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
If P = Q, then we can stilladd points.
Here, we use the tangentline to find a third point ofintersection.
To the right, we start withthe point P = (−1, 1) onthe same elliptic curve.
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
Using calculus, we cancalculate the tangent line atP to be y = x + 2.
This intersects the ellipticcurve at the point (3, 5).
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
Reflecting as before gives usthat 2P = P + P = (3,−5).
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve
What about if the linebetween P and Q is vertical?
We define a “point atinfinity” and call it R = O.This point intersects allvertical lines.
In this case, we also callQ = −P (this is thereflection of P about thex-axis).
ThusP − P = P + Q = R = O
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Your Turn!
Try an example. Add thepoints P = (0, 1) andQ = (3, 5).
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Your Turn!
The slope of the linebetween P and Q is
m =5− 1
3− 0=
4
3
and the y intercept is b = 1since P = (0, 1) is on theline y = 4
3x + 1.
Thus the equation of theline between P and Q isy = 4
3x + 1.
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Your Turn!
Where does the line y = 43x + 1 intersect y2 = x3 − x + 1?
Plug the equation of the line into the elliptic curve to get:
(43x + 1)2 = x3 − x + 1169 x
2 + 83x + 1 = x3 − x + 1
x3 − 169 x
2 − 113 x = 0
x(x2 − 169 x −
113 ) = 0
The last quadratic must have x = 3 as a root since we knowthe line intersects at the points P = (0, 1) and Q = (3, 5). Sofactoring the above gives
x(x − 3)(x + 119 ) = 0
Thus the other point of intersection occurs when x = −119 .
The corresponding y value is
y = 43(−11
9 ) + 1 = −4427 + 1 = −17
27
Dr. Carmen Bruni Elliptic Curves
Your Turn!
This line intersects theelliptic curve at the point(−11
9 , −1727 ).
Then finally, reflecting(negating the y -coordinate)gives the pointR = (−11
9 , 1727)
y2 = x3 − x + 1
Dr. Carmen Bruni Elliptic Curves
Formulas For Adding Points
Let’s summarize the above for adding two points P = (x1, y1)and Q = (x2, y2) on the elliptic curve y2 = x3 + Cx + D.
Let ` be the line connecting P and Q and suppose ` is definedby
y = mx + b
We can describe the slope m and the y -intercept b via
m =
{y2−y1x2−x1
If P 6= Q3x21+C2y1
If P = Qand b = y1 −mx1
where again we used calculus to compute the tangent line inthe case when P = Q.
Dr. Carmen Bruni Elliptic Curves
Formulas For Adding Points
As in our example, we can find the intersection of
y2 = x3 + Cx + D and y = mx + b
by solving
(mx + b)2 = x3 + Cx + D
m2x2 + 2mxb + b2 = x3 + Cx + D
0 = x3 −m2x2 + (C − 2mb)x + D − b2
Dr. Carmen Bruni Elliptic Curves
Formulas For Adding Points
This new polynomial has x1 and x2 as solutions since P and Q areon both the line and the curve. Hence,
0 = x3 −m2x2 + (C − 2mb)x + D − b2
= (x − x1)(x − x2)(x − x3)
= x3 − (x1 + x2 + x3)x2 + (x1x2 + x1x3 + x2x3)x − x1x2x3
which must hold for all values of x . Hence the coefficients oneither side match up. Thus, comparing the x2 coefficients oneither side gives
−m2 = −(x1 + x2 + x3) ⇒ x3 = m2 − x1 − x2
and y3 = mx3 + b. Hence reflecting gives
P + Q = (x3,−y3)
Dr. Carmen Bruni Elliptic Curves
Formulas For Adding Points on y 2 = x3 − N2x
When we add P = (x , y) = Q on the elliptic curvey2 = x3 − N2x with N squarefree, the formula for thex-coordinate of P + P becomes:
(x2 − N2)2
(2y)2
(see the problem set). Notice here that the x-coordinate is asquare, has an even denominator and the numerator shares nocommon factor with N provided P 6= (0, 0) or (±N, 0) (seethe problem set).
Dr. Carmen Bruni Elliptic Curves
Revisit the Congruent Number Problem
Congruent Number Problem
Determine which positive integers N can be expressed as the areaof a right angled triangle with side lengths all rational.
Dr. Carmen Bruni Elliptic Curves
Key Theorem 1
Theorem 1.
Let (x , y) be a point with rational coordinates on the elliptic curvey2 = x3 − N2x where N is a positive squarefree integer. Supposethat x satisfies three conditions:
1 x is the square of a rational number
2 x has an even denominator
3 x has a numerator that shares no common factor with N
Then there exists a right angle triangle with rational sides and areaN, that is, N is congruent.
Dr. Carmen Bruni Elliptic Curves
Key Theorem 2
Theorem 2.
A number N is congruent if and only if the elliptic curvey2 = x3 − N2x has a rational point P = (x , y) distinct from (0, 0)and (±N, 0).
Thus, determining congruent numbers can be reduced to findingrational points on elliptic curves!
Dr. Carmen Bruni Elliptic Curves
Next Time
We prove these theorems.
We figure out how to go from a rational point on an ellipticcurve to a rational right triangle with area N.
We revisit Don Zagier’s example.
We discuss some tricks for finding rational points on ellipticcurves.
Dr. Carmen Bruni Elliptic Curves