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G13ELL=MATH3031 Elliptic Curves G13ELL=MATH3031 Elliptic Curves 2018–19 Chris Wuthrich Elliptic curves G13ELL=MATH3031 cw ’18 Essential information Lecturer : Chris Wuthrich, [email protected], phone 14920. Lectures : • Tuesdays 10 am – noon, Physics C29 • Thursdays noon – 1pm, Pharmacy A6 • Fridays 11 am – noon, Maths A17 Office Hours : On Thursday mornings in my office C58 in the Maths build- ing. If you wish to meet me at any other time, just try knocking at the door or – if you want to make certain that I am in – please contact me by email or by phone. Module webpage : The module webpage is on moodle: http://moodle.nottingham.ac.uk/course/view.php?id=68482. Lecture notes : On the moodle page you will find all these lecture notes including all the pictures. The printed version omits pictures that you are asked to draw during lectures. Booklist : There are plenty of books and online lecture material on elliptic curves. This module recommends [6], [1] and [7] (in the list on page 4) as the best books to consult. Please ask if you are interested in a particular topic. Problem classes : We will have problem classes, in average one per week. During this hour you will work (with my help) on exercises relating to the lectures. There are also unassessed coursework sheets which you are asked to hand in your solutions for marking. Assessment : There is a 3-hour exam in January. There will be FIVE ques- tions in the exam paper. Credit will be given for the best FOUR answers. Consult the moodle page for the exam paper of last year. More information will be given later. Computer software : Not needed at all, but if you wish to experiment with elliptic curves you may want to try out the free SageMath. You may use it freely on CoCalc. All graphics and computations in these notes are done with SageMath. 1 Elliptic curves G13ELL=MATH3031 cw ’18 Contents 1 Chords and tangents 5 2 Groups and fields 10 2.1 Abelian groups . 10 2.2 Fields . 11 3 The projective line and plane 13 3.1 A motivation . 13 3.2 Algebraic definition . 13 1 3.3 The projective line P ....................... 14 2 3.4 The projective plane P ...................... 15 3.5 Parametrising lines . 18 4 Projective curves 20 4.1 Conics . 23 4.2 Multiplicities and singularities . 28 4.3 Bézout’s theorem . 35 5 Elliptic Curves 38 5.1 The group law . 40 5.2 Formula on Weierstrass equations . 45 5.3 Reduction modulo p ........................ 50 5.4 Transforming a cubic into a Weierstrass equation . 51 6 Torsion points 55 6.1 Points of order two . 55 6.2 Torsion points have integer coordinates . 57 6.3 The theorem of Lutz and Nagell . 61 6.4 Mazur’s theorem . 62 7 Mordell’s theorem 63 7.1 The weak theorem . 63 7.2 Heights . 68 7.3 Determination of the rank . 71 8 Additional topics 75 8.1 History . 75 8.2 Cryptography . 77 8.3 Primality testing . 79 2 Elliptic curves G13ELL=MATH3031 cw ’18 8.4 Factorisation . 80 8.5 Elliptic curves over C ........................ 82 8.6 Birch and Swinnerton-Dyer conjecture . 84 8.7 Fermat’s Last Theorem . 86 3 Elliptic curves G13ELL=MATH3031 cw ’18 References [1] J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Soci- ety Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991, George Green Library, QA565 CAS. [2] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert, Numbers, Graduate Texts in Mathematics, vol. 123, Springer-Verlag, New York, 1991. [3] David Eisenbud, Mark Green, and Joe Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295–324. [4] Kenneth Ireland and Michael Rosen, A classical introduction to mod- ern number theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. [5] Franz Lemmermayer, Elliptic curves, historical remarks, Avaialble at http: //www.fen.bilkent.edu.tr/~franz/ta/ta01.pdf, last visited Sep 2018. [6] Joseph H. Silverman and John T. Tate, Rational points on elliptic curves, second ed., Undergraduate Texts in Mathematics, Springer, Cham, 2015, George Green Library, QA565 SIL. [7] Lawrence C. Washington, Elliptic curves, second ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008, George Green Library, QA565 WAS and QA567.2.E44 WAS. 4 Elliptic curves G13ELL=MATH3031 cw ’18 1 Chords and tangents We start this module on elliptic curves with some examples. The motto here is “Geometry helps solving equations”. Say, we want to solve the equation C : x2 + y2 − 2x = 0; (1.1) however if possible without using square roots or non-algebraic func- tions like sin and log. We start by making a drawing as in Figure 1. Since the equation can be reformulated as (x − 1)2 + y2 = 1 we see a circle of ra- dius 1 centred at (1; 0). For instance P = (0; 0) is an obvious solution. Pass a line through P . It can be given by the equation Figure 1: A geometric picture of our L : y = mx equation (1.1) where m is some constant. The points in the intersection L \ C satisfy x2 + (mx)2 − 2x = 0 ) (1 + m2) x2 − 2x = 0 ) x · (1 + m2) x − 2 = 0 If x = 0, then y = m · 0 = 0 and we get back the point P . Of course this is no surprise as P was on C and L. Otherwise, if x 6= 0, we have 2 2m Q = (x; y) = ; : (1.2) 1 + m2 1 + m2 Is Q really on C? Well, . 2 2 2m 2 2 2 + 4m2 − 2(1 + m2) + − 2 = = 0 (1.3) 1 + m2 1 + m2 1 + m2 (1 + m2)2 so yes, Q is in C \ L. Therefore we found plenty of solutions to C, but did we find all of them? Well, if (x0; y0) is a solution to C with x0 6= 0, then there is an m = y0=x0 such that the line y = mx passes through P and (x0; y0). Therefore we will recover any solution to C other than P from the parametrisation (1.2). We can write 2 2m n o C(R) = ; m 2 R [ (0; 0) 1 + m2 1 + m2 5 Elliptic curves G13ELL=MATH3031 cw ’18 where we have denoted by C(R) the solution set for C with coordinates x and y in R. The real numbers? Yes, when we relied on our picture we really made a picture for the real solutions. But what if we would like to find all solutions to (1.1) with coordinates in k = Q or k = C. Does the above still work? First of all the computation (1.3) works for any field k and m 2 k as long as m2 6= −1. So no matter what k is, we find a parametrisation. For any solution (x0; y0) with 0 6= x0 2 k and y0 2 k we can set m = y0=x0 and thus we recover all solutions except when m2 = −1. For instance we get that 2 2m n o C(Q) = ; m 2 Q [ (0; 0) : 1 + m2 1 + m2 It even works for k = F3 = Z=3Z, the field of three elements: 2 2m n o C(F3) = ; m 2 F3 [ (0; 0) 1 + m2 1 + m2 n o = (2; 0); (1; 1); (1; 2); (0; 0) : Figure 2: The “circle” (1.1) over F3 and over F17 However this time the picture is quite far from a circle. See Figure 2. How- ever the geometric method helped us to find all solutions quicker than if we checked all possible x and y in F3. We pass to a more complicated example. In the above the equation was of degree 2 and we will later see that the method used there generalises. Our next example is of degree 3, it will turn out to be an example of an elliptic curve. The equation is C : x3 + y3 = 1729 (1.4) 6 Elliptic curves G13ELL=MATH3031 cw ’18 and we are looking for solutions with coordinates in Q. One may be able to spot∗ some solutions, including P = (1; 12) and Q = (9; 10). See Figure 3. Let Figure 3: Chords on the taxicab curve us pick a general line through P L : y − 12 = m(x − 1) for some m 2 Q. The points in the intersection C \ L satisfy 0 = x3 + (mx − m + 12)3 − 1729 = (1 + m3)x3 + 3m2(12 − m)x2 + 3m(m2 − 24m + 144)x − m3 + 36m2 − 432m − 1 = (x − 1)(1 + m2)x2 − (2m3 − 36m2 − 1)x + m3 − 36m2 + 432m + 1 where, in the last line, we found a factorisation because we knew that x = 1 must be a solution, given that P 2 C \ L. Now we are stuck as we are left with a quadratic polynomial that does not seem to factor. All is not lost. Let L be the line through P and Q. This is a particular line 1 for which we know a second solution. The line corresponds to m = − 4 above. We find 3 x−1 3 21 0 = x + − 4 + 12 − 1729 = 64 (x − 1)(x − 9)(3x + 37): ∗1729 is a famous taxicab number. Hardy visited Ramanujan in 1919. He wrote: “I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen.
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