The Ubiquity of Elliptic Curves Joseph Silverman (Brown University) Public Lecture – Dublin Tuesday, 4 September 2007, 7:30 PM

Elliptic Curves Geometry, Algebra, Analysis and Beyond…

An elliptic curve is an object with a dual nature: On the one hand, it is a curve, a geometric object. On the other hand, we can “add” points on the curve as if they were numbers, so it is an algebraic object. The addition law on an elliptic curve can be described: Geometricallyusing intersections of curves Algebraicallyusing polynomial equations Analyticallyusing functions with complex variables Elliptic curves appear in many diverse areas of mathematics, ranging from number theory to complex analysis, and from cryptography to mathematical physics. What is an Elliptic Curve? - 3 -

The Equation of an Elliptic Curve An Elliptic Curve is a curve given by an equation E : y 2 = f(x) for a cubic or quartic polynomial f(x) We also require that the polynomial f(x) has no double roots. This ensures that the curve is nonsingular After a change of variables, the equation takes the simpler form E : y 2 = x 3 + A x + B Finally, for reasons to be explained shortly, we toss in an extra point O “at infinity,” so E is really the set E = { (x,y) : y 2 = x 3 + A x + B }  { O }

A Typical Elliptic Curve E E : Y 2 = X 3 – 5X Surprising Fact: We can use geometry to take two points P and Q on the elliptic curve and define their “sum” P+Q.

The Addition Law on an Elliptic Curve

Adding Points P + Q on E P Q P+Q R - 7 -

Doubling a Point P on E P 2*P R Tangent Line to E at P - 8 -

Vertical Lines and an Extra Point at Infinity Vertical lines have no third intersection point Q Add an extra point O “at infinity.” The point O lies on every vertical line. O PQ = –P - 9 -

Properties of “Addition” on E Theorem: The addition law on E has the following properties: a)P + O = O + P = Pfor all P  E. b)P + (–P) = O for all P  E. c)(P + Q) + R = P + (Q + R)for all P,Q,R  E. d)P + Q = Q + Pfor all P,Q  E. In mathematical terminology, the addition law + makes the points of E into a commutative group. All of the group properties are easy to check except for the associative law (c). The associative law can be verified by a lengthy computation using explicit formulas, or by using more advanced algebraic or analytic methods

An Example Using the tangent line construction, we find that 2P = P + P = (– 7/4, – 27/8). Using the secant line construction, we find that 3P = P + P + P = (553/121, – 11950/1331) Similarly, 4P = (45313/11664, / ). As you can see, the coordinates become complicated. E : Y 2 = X 3 – 5X + 8 The point P = (1,2) is on the curve E

An Addition Formula for E Suppose that we want to add the points P 1 = (x 1,y 1 ) and P 2 = (x 2,y 2 ) on the elliptic curve E : y 2 = x 3 + Ax + B Quite a mess!!!!! But… Crucial Observation: If A and B are rational numbers and if the coordinates of P 1 and P 2 are rational numbers, then the coordinates P 1 + P 2 and 2P 1 are rational numbers.

The Group of Points on E with Rational Coordinates The elementary observation on the previous slide leads to an important result: Theorem (Poincaré,  1900): Suppose that an elliptic curve E is given by an equation of the form y 2 = x 3 + A x + B with A,B rational numbers. Let E(Q) be the set of points of E with rational coordinates, E(Q) = { (x,y)  E : x,y are rational numbers }  { O }. Then sums of points in E(Q) remain in E(Q) In mathematical terminology, E(Q) is a subgroup of E.

The Group of Points on E with Other Sort of Coordinates In an earlier slide, we drew a picture of E in the plane. This was the set of points of E with real coordinates: E(R) = { (x,y)  E : x,y are real numbers }  { O } Similarly, we can look at the points of E whose coordinates are complex numbers: E(C) = { (x,y)  E : x,y are complex numbers }  { O }. And later we’ll look at the set of points E(F p ) whose coordinates are in a “finite field” F p. Key Fact: In any of these sets, we can add points and stay within the set.

What Does E(R) Look Like? We saw one example of E(R). It is also possible for E(R) to have two connected components E : Y 2 = X 3 – 9X

Elliptic Curves and Complex Numbers Or…How the Elliptic Curve Acquired Its Unfortunate Moniker

The Arc Length of an Ellipse The arc length of a half circle -aa x 2 +y 2 =a 2 is given by the familiar integral is more complicated The arc length of a half ellipse x 2 /a 2 + y 2 /b 2 = 1 -a b a

An Elliptic Curve! The Arc Length of an Ellipse Let k 2 = 1 – b 2 /a 2 and change variables x  ax. Then the arc length of an ellipse is with y 2 = (1 – x 2 ) (1 – k 2 x 2 ) = quartic in x. An elliptic integral is an integral, where R(x,y) is a rational function of the coordinates (x,y) on an “elliptic curve” E : y 2 = f(x) = cubic or quartic in x.

Elliptic Integrals and Elliptic Functions Doubly periodic functions are called elliptic functions. Its inverse function w = sin(z) is periodic with period 2 . The circular integral is equal to sin -1 (w). The elliptic integral has an inverse w =  (z) with two independent complex periods  1 and  2.  (z +  1 ) =  (z +  2 ) =  (z) for all z  C.

Elliptic Functions and Elliptic Curves This equation looks familiar The  -function and its derivative satisfy an algebraic relation The double periodicity of  (z) means that it is a function on the quotient space C/L, where L is the lattice L = { n 1  1 + n 2  2 : n 1,n 2  Z }. 11 22  1 +  2 L  (z) and  ’(z) are functions on a fundamental parallelogram

The Complex Points on an Elliptic Curve E(C) = The  -function gives a complex analytic isomorphism Thus the points of E with coordinates in the complex numbers C form a torus, that is, the surface of a donut. E(C) Parallelogram with opposite sides identified = a torus

Elliptic Curves and Number Theory Rational Points on Elliptic Curves

E(Q) : The Group of Rational Points A fundamental and ancient problem in number theory is that of solving polynomial equations using integers or rational numbers. The description of E(Q) is a landmark in the modern study of Diophantine equations. Theorem (Mordell, 1922): Let E be an elliptic curve given by an equation E : y 2 = x 3 + A x + B with A,B  Q. There is a finite set of points P 1,P 2,…,P r so that every point P in E(Q) can be obtained as a sum P = n 1 P 1 + n 2 P 2 + … + n r P r with n 1,…,n r  Z. In math terms, E(Q) is a finitely generated group

E(Q) : The Group of Rational Points A point P has finite order if some multiple of P is O. The elements of finite order in E(Q) are quite well understood Theorem (Mazur, 1977): The group E(Q) contains at most 16 points of finite order. Conjecture: The number of points needed to generate E(Q) may be arbitrarily large. The minimal number of points needed to generate the group E(Q) is much more mysterious! Current World Record: (Elkies 2006) There is an elliptic curve with Number of generators for E(Q)  28.

E(Z) : The Set of Integer Points If P 1 and P 2 are points on E having integer coordinates, then P 1 + P 2 will have rational coordinates, but there is no reason for it to have integer coordinates. Indeed, the formulas for P 1 + P 2 are so complicated, it seems unlikely that P 1 + P 2 will have integer coordinates. Complementing Mordell’s finite generation theorem for rational points is a famous finiteness result for integer points Theorem (Siegel, 1928): An elliptic curve E : y 2 = x 3 + A x + B with A,B  Z has only finitely many points P = (x,y) with integer coordinates x,y  Z.

Elliptic Curves and Finite Fields

Finite Fields You may have run across clock arithmetic, where after counting 0, 1, 2, 3,…,11, you go back to Another way to view clock arithmetic is that whenever you add or multiply numbers together, you should divide by 12 and just keep the remainder. We want to do the same thing, but instead of using 12, we’ll use a prime number p, for example 3 or 7 or 37. The Finite Field F p is the set of numbers 0, 1, 2, …, p–1 with the rule that when we add or multiply two of them, we are required to divide by p and just keep the remainder.

An Example of a Finite Field For example, in the finite field F 7, since 9 divided by 7 leaves remainder 2. since 20 divided by 7 leaves remainder 6. Similarly, in the field F 41, we have 11 x 15 = 1 and 23 x 25 = 1 and 19 x 13 = 1 and … This illustrates why we use a prime p, instead of a number like 12. In a finite field F p, every nonzero number has a reciprocal. So F p is a lot like the rational numbers Q and the real numbers R: In F p, not only can we can add, subtract, and multiply, we can also divide by nonzero numbers

Elliptic Curves over a Finite Field The formulas giving the addition law on E are fine if the points have coordinates in any field, even if the geometric pictures don’t make sense. For example, we can take points with coordinates in F p. Example:The curve E : Y 2 = X 3 – 5X + 8 modulo 37 contains the points P = (6,3) and Q = (9,10). Using the addition formulas, we can compute in E( F 37 ): 2P = (35,11) 3P = (34,25) 4P = (8,6) 5P = (16,19) … P + Q = (11,10) 3P + 4Q = (31,28) …

E(F p ) : The Group of Points Modulo p Number theorists also like to solve polynomial equations modulo p Theorem (Hasse, 1922): An elliptic curve equation E : y 2  x 3 + A x + B (modulo p) has p  solutions (x,y) mod p, where the error  satisfies This is much easier than finding solutions in Q, since there are only finitely many solutions in the finite field F p ! One expects E(F p ) to have approximately p+1 points. A famous theorem of Hasse (later vastly generalized by Weil and Deligne) quantifies this expectation.

Elliptic Curves and Cryptography

The (Elliptic Curve) Discrete Log Problem Suppose that you are given two points P and Q in E(F p ) If the prime p is large, it is very very difficult to find m. Neal Koblitz and Victor Miller (1985) independently invented Elliptic Curve Cryptography in 1985 when they suggested building a cryptosystem around the ECDLP. The extreme difficulty of the ECDLP yields highly efficient cryptosystems that are in widespread use protecting everything from your bank account to your government’s secrets. The Elliptic Curve Discrete Logarithm Problem (ECDLP) is to find an integer m satisfying Q = P + P + … + P = mP. m summands

Elliptic Curve Diffie-Hellman Key Exchange Public Knowledge: A group E(F p ) and a point P of order n. BOB ALICE Choose secret 0 < b < n Choose secret 0 < a < n Compute Q Bob = bP Compute Q Alice = aP Compute bQ Alice Compute aQ Bob Bob and Alice have the shared value bQ Alice = abP = aQ Bob Presumably(?) recovering abP from aP and bP requires solving the elliptic curve discrete logarithm problem. Send Q Bob to Alice to Bob Send Q Alice

Elliptic Curves and Classical Physics

The Pit and the Pendulum

The Pit and the Pendulum This leads to a simple harmonic motion for the pendulum. In freshman physics, one assumes that  is small and derives the formula But this formula is only a rough approximation. The actual differential equation for the pendulum is

How to Solve the Pendulum Equation Conclusion: tan(  /2) = Elliptic Function of t An Elliptic Curve!!! An Elliptic Integral!!! and do a bunch of algebra. As a favor, I’ll spare you the details and just tell you the answer!! To solve the pendulum equation, we make the substitution

Elliptic Curves and Modern Physics

Elliptic Curves and String Theory In string theory, the notion of a point-like particle is replaced by a curve-like string. As a string moves through space-time, it traces out a surface. For example, a single string that moves around and returns to its starting position will trace a torus. So the path traced by a string looks like an elliptic curve! In quantum theory, physicists like to compute averages over all possible paths, so when using strings, they need to compute integrals over the space of all elliptic curves.

Elliptic Curves and Number Theory Fermat’s Last Theorem

Fermat’s Last Theorem and Fermat Curves Fermat’s Last Theorem says that if n > 2, then the equation a n + b n = c n has no solutions in nonzero integers a,b,c. It is enough to prove the case that n = 4 (already done by Fermat himself) and the case that n = p is an odd prime. If we let x = a/c and y = b/c, then solutions to Fermat’s equation give rational points on the Fermat curve x p + y p = 1. But Fermat’s curve is not an elliptic curve. So how can elliptic curves be used to study Fermat’s problem?

Elliptic Curves and Fermat’s Last Theorem Frey suggested that E a,b,c would be such a strange curve, it shouldn’t exist at all. More precisely, Frey doubted that E a,b,c could be modular. Ribet verified Frey’s intuition by proving that E a,b,c is indeed not modular. Wiles completed the proof of Fermat’s Last Theorem by showing that (most) elliptic curves, in particular elliptic curves like E a,b,c, are modular. Gerhard Frey (and others) suggested using an hypothetical solution (a,b,c) of Fermat’s equation to “manufacture” an elliptic curve E a,b,c : y 2 = x (x – a p ) (x + b p ).

Elliptic Curves and Fermat’s Last Theorem To Summarize: Suppose that a p + b p = c p with abc  0. Ribet proved that E a,b,c is not modular Wiles proved that E a,b,c is modular. Conclusion: The equation a p + b p = c p has no solutions. E a,b,c : y 2 = x (x – a p ) (x + b p ) But what does it mean for an elliptic curve E to be modular?

The variable  represents the elliptic curve E  whose lattice is L  = {n 1 +n 2  : n 1,n 2  Z }. So just as in string theory, the space of all elliptic curves makes an unexpected appearance. Elliptic Curves and Modularity E is modular if it is parameterized by modular forms! There are many equivalent definitions, all of them rather complicated and technical. Here’s one: A modular form is a function f(  ) with the property

Conclusion

The Ubiquity of Elliptic Curves Joseph Silverman (Brown University) Public Lecture Dublin – September 4, 2007