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 area of a right angled triangle with side lengths all rational. For example 6 is a congruent number since it is the area of the 3 − 4 − 5 right triangle. Dr. Carmen Bruni Elliptic Curves
From Triangles to Curves Now, we’re going to take the information about our triangle and get a new equation which will turn out to represent a curve in the real plane. Let x 2 + y 2 = z 2 and xy = 2 N for rationals x , y , z and some congruent number N . Adding and subtracting 2 xy = 4 N to the first equation gives x 2 + 2 xy + y 2 = z 2 + 4 N x 2 − 2 xy + y 2 = z 2 − 4 N Factoring and dividing by 4 gives the two equations � 2 � 2 � x + y � x − y = ( z / 2) 2 + N = ( z / 2) 2 − N 2 2 Dr. Carmen Bruni Elliptic Curves
From Triangles to Curves With the equations � 2 � 2 � x + y � x − y = ( z / 2) 2 + N = ( z / 2) 2 − N 2 2 we multiply these two equations together gives � 2 � 2 � x + y � x − y = (( z / 2) 2 + N )(( z / 2) 2 − N ) · 2 2 �� 2 �� x + y � � x − y = ( z / 2) 4 + N ( z / 2) 2 − N ( z / 2) 2 − N 2 · 2 2 � 2 � ( x − y )( x + y ) = ( z / 2) 4 − N 2 4 � x 2 − y 2 � 2 = ( z / 2) 4 − N 2 4 Dr. Carmen Bruni Elliptic Curves
From Triangles to Curves Letting u = z / 2 and v = ( x 2 − y 2 ) / 4, the previous equation becomes v 2 = u 4 − N 2 Multiplying by u 2 gives ( uv ) 2 = ( u 2 ) 3 − N 2 u 2 Finally, we let y = uv and x = u 2 which gives us the equation y 2 = x 3 − N 2 x We call such curves where y 2 equals a cubic in x an Elliptic Curve (provided the discriminant is nonzero; this is the case for 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 look like on the real plane? Let’s try to draw y 2 = x 3 − x first by drawing y = x 3 − x and then trying to draw the elliptic curve. Dr. Carmen Bruni Elliptic Curves
Drawing y = x 3 − x First, note that y = x 3 − x = x ( x − 1)( x + 1) and so the equation has three zeroes at x = 0 , ± 1. Now let’s break this curve into four intervals and see what happens in each interval y = x 3 − 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 = x 3 − x Here is the picture (Using Desmos.com) Dr. Carmen Bruni Elliptic Curves
The Elliptic Curve y 2 = x 3 − x What changes when we make the left hand side y 2 instead of y ? For almost all values of x , we will get not 1 but 2 output values (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 the areas where the picture is negative are gone. The curve still has no breaks and is symmetric about the x -axis, that is, if I reflect the top half of the picture, it should match the bottom half. The function should still be smooth (even at 1). Dr. Carmen Bruni Elliptic Curves
The Cubic Curve y = x 3 − 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 y 2 = x 3 − 1: However, the elliptic curves associated to the Congruent Number Problem 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, y 2 = x 3 − x has infinitely many real points by noticing that the cubic on the right is always positive when x > 1 and hence we can find a y value by taking the square root. So if we take x = 2, then we see that y 2 = 2 3 − 2 = 6 and so √ √ the 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 is interesting to ask: How many integer points are on elliptic curves? For the example y 2 = x 3 − x , it turns out that ( ± 1 , 0) and (0 , 0) are the only integer points, though this is hardly obvious. How many rational points are on elliptic curves? Above, the only rational points are also the integral ones. More on this later. Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 With an elliptic curve, we can actually describe a way to, given two rational points P and Q , create a third rational point R . Let’s begin with the elliptic curve y 2 = x 3 − x + 1 for illustrative purposes. Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 Let’s take the points P = ( − 1 . 324 , 0) and Q = (0 , 1) (correct to three decimal places). Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 Draw the line between P and Q . It intersects the curve in a third point as shown in the picture at coordinates (1 . 895 , 2 . 43). Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 Draw the vertical line through the point which must intersect the curve in a third point, in our case, R = (1 . 895 , − 2 . 43) (this is the same as reflecting about the x-axis). Define P + Q = R for points on an elliptic curve (note that this isn’t just adding the coordinates!) Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 If P = Q , then we can still add points. Here, we use the tangent line to find a third point of intersection. To the right, we start with the point P = ( − 1 , 1) on the same elliptic curve. Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 Using calculus, we can calculate the tangent line at P to be y = x + 2. This intersects the elliptic curve at the point (3 , 5). Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 Reflecting as before gives us that 2 P = P + P = (3 , − 5). Dr. Carmen Bruni Elliptic Curves
Group Law of an Elliptic Curve y 2 = x 3 − x + 1 What about if the line between P and Q is vertical? We define a “point at infinity” and call it R = O . This point intersects all vertical lines. In this case, we also call Q = − P (this is the reflection of P about the x -axis). Thus P − P = P + Q = R = O Dr. Carmen Bruni Elliptic Curves
Your Turn! y 2 = x 3 − x + 1 Try an example. Add the points P = (0 , 1) and Q = (3 , 5). Dr. Carmen Bruni Elliptic Curves
Your Turn! y 2 = x 3 − x + 1 The slope of the line between P and Q is m = 5 − 1 3 − 0 = 4 3 and the y intercept is b = 1 since P = (0 , 1) is on the line y = 4 3 x + 1. Thus the equation of the line between P and Q is y = 4 3 x + 1. Dr. Carmen Bruni Elliptic Curves
Your Turn! 3 x + 1 intersect y 2 = x 3 − x + 1? Where does the line y = 4 Plug the equation of the line into the elliptic curve to get: 3 x + 1) 2 = x 3 − x + 1 ( 4 9 x 2 + 8 3 x + 1 = x 3 − x + 1 16 x 3 − 16 9 x 2 − 11 3 x = 0 x ( x 2 − 16 9 x − 11 3 ) = 0 The last quadratic must have x = 3 as a root since we know the line intersects at the points P = (0 , 1) and Q = (3 , 5). So factoring the above gives x ( x − 3)( x + 11 9 ) = 0 Thus the other point of intersection occurs when x = − 11 9 . The corresponding y value is y = 4 3 ( − 11 9 ) + 1 = − 44 27 + 1 = − 17 27 Dr. Carmen Bruni Elliptic Curves
Your Turn! y 2 = x 3 − x + 1 This line intersects the elliptic curve at the point ( − 11 9 , − 17 27 ). Then finally, reflecting (negating the y -coordinate) gives the point R = ( − 11 9 , 17 27 ) Dr. Carmen Bruni Elliptic Curves
Formulas For Adding Points Let’s summarize the above for adding two points P = ( x 1 , y 1 ) and Q = ( x 2 , y 2 ) on the elliptic curve y 2 = x 3 + Cx + D . Let ℓ be the line connecting P and Q and suppose ℓ is defined by y = mx + b We can describe the slope m and the y -intercept b via � y 2 − y 1 If P � = Q x 2 − x 1 m = and b = y 1 − mx 1 3 x 2 1 + C If P = Q 2 y 1 where again we used calculus to compute the tangent line in the case when P = Q . Dr. Carmen Bruni Elliptic Curves
Formulas For Adding Points As in our example, we can find the intersection of y 2 = x 3 + Cx + D and y = mx + b by solving ( mx + b ) 2 = x 3 + Cx + D m 2 x 2 + 2 mxb + b 2 = x 3 + Cx + D 0 = x 3 − m 2 x 2 + ( C − 2 mb ) x + D − b 2 Dr. Carmen Bruni Elliptic Curves
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