# Right triangles and elliptic curves

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1 Right triangles and elliptic curves Karl Rubin Ross Reunion July 2007 Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

2 Rational right triangles Question Given a positive integer d, is there a right triangle with rational sides and area d? Pythagorean Theorem: A d C Area: d = AB 2 A 2 + B 2 = C 2 B Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

3 Rational right triangles Question Given a positive integer d, is there a right triangle with rational sides and area d? Examples: A = 3, B = 4, C = 5 A d C d = 6 A = 3, B = 20, C = d = 5 B Theorem (Fermat, 1640) There is no rational right triangle with area 1. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

4 Answer Suppose d is a positive integer, not divisible by the square of an integer bigger than 1. Let a = 1 if d is odd, and a = 2 if d is even, and n = #{(x, y, z) Z 3 : x 2 + 2ay 2 + 8z 2 = d/a} m = #{(x, y, z) Z 3 : x 2 + 2ay z 2 = d/a} counting integer solutions (positive, negative, or zero) x, y, z. Theorem (Tunnell, 1983) If n 2m, then there is no rational right triangle with area d. Conjecture If n = 2m, then there is a rational right triangle with area d. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

5 Answer Suppose d is a positive squarefree integer, and a = (d, 2). Let n = #{(x, y, z) : x 2 + 2ay 2 + 8z 2 = d/a} m = #{(x, y, z) : x 2 + 2ay z 2 = d/a} d , 6, or 7 (mod 8) n m Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

6 Answer d right triangle with area d 1 none 2 none 3 none 5 (3/2, 20/3, 41/6) 6 (3, 4, 5) 7 (24/5, 35/12, 337/60) 11 none 41 (40/3, 123/20, 881/60) 157? d , 6, or 7 (mod 8) n m Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

7 d = 157 A Area = 157 C B A = B = C = Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

8 5, 6, and 7 (mod 8) Conjecture If d is positive, squarefree, and d 5, 6, or 7 (mod 8), then there is a rational right triangle with area d. This has been verified for d < 1, 000, 000. Theorem If p is a prime, and p 5 or 7 (mod 8), then there is a rational right triangle with area p. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

9 Translating the question C If A has area d... B... then x = 1 2 A(A C), y = 1 2 A2 (C A) is a solution of y 2 = x 3 d 2 x. For example, the (3, 4, 5) triangle with area 6 gives the solution ( 3, 9) of y 2 = x 3 36x. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

10 Translating the question If (x, y) is a solution of y 2 = x 3 d 2 x, and y 0... then x2 d 2 y x2 +d 2 y is a right triangle with area d. 2xd y Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

11 Translating the question Theorem There is a rational right triangle with area d if and only if there are rational numbers x and y, y 0, such that y 2 = x 3 d 2 x. The equation y 2 = x 3 d 2 x is an elliptic curve. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

12 y 2 = x 3 36x P ( 3, 9) ( , ) ( 25 4, 35 8 ) ( 25 4, 35 8 ) P = ( , ) Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

13 y 2 = x 3 36x In fact, this procedure gives infinitely many rational solutions (x, y) of the equation y 2 = x 3 36x, so there are infinitely many rational right triangles with area 6. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

14 Some right triangles with area have about.38n 2 digits The numerator and denominator of a side of the n-th triangle each Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

15 Elliptic curves An elliptic curve is a curve defined by a cubic equation y 2 = x 3 + ax + b with constants a, b Z, and := 16(4a b 2 ) 0. (One should really think of it as a curve Y 2 Z = X 3 + axz 2 + bz 3 in 2-dimensional projective space.) Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

16 Elliptic curves Basic Problem If E is the elliptic curve y 2 = x 3 + ax + b, find all rational solutions (rational points): E(Q) := {(x, y) Q Q : y 2 = x 3 + ax + b} { } Example (Fermat) If E is y 2 = x 3 x, then E(Q) = {(0, 0), (1, 0), ( 1, 0), }. (In particular, there is no rational right triangle with area 1.) The chord-and-tangent process can be used to define an addition law on E(Q), making E(Q) a commutative group. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

17 Elliptic curves (P + Q) Q P P + Q Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

18 Elliptic curves Example (Fermat) If E is y 2 = x 3 x, then E(Q) = Z 2 Z 2. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

19 Elliptic curves Theorem (Mordell, 1922) The group E(Q) is finitely generated. In other words, although E(Q) may be infinite, there is always a finite set of points {P 1,..., P r } that generates all points in E(Q) under the chord-and-tangent process. In other other words, where E(Q) = Z r E(Q) tors r is a nonnegative integer, called the rank of E, E(Q) tors is a finite group, made up of all points that have finite order under the group law on E. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

20 Points of finite order Theorem (Nagell, Lutz, 1937) If (x, y) E(Q) tors, then x, y Z and either y = 0 or y 2. Theorem (Mazur, 1977) E(Q) tors is one of the following 15 groups: Z n, 1 n 10 or n = 12, Z 2 Z 2m, 1 m 4 and each of these groups occurs infinitely often. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

21 Points of finite order Example If E d is y 2 = x 3 d 2 x then E d (Q) tors = {(0, 0), (d, 0), ( d, 0), } = Z 2 Z 2. Theorem There is a rational right triangle with area d if and only if E d (Q) is infinite. Corollary If there is one rational right triangle with area d, then there are infinitely many. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

22 Ranks Equivalently, there is a rational right triangle with area d if and only if the rank of E(Q) is nonzero. Unfortunately, the rank is very mysterious. There is no known algorithm guaranteed to determine the rank. It is not known which ranks can occur. How can we determine the rank, or at least determine whether E(Q) is infinite? Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

23 Ranks Another interpretation of the rank: There is a constant C R + such that #{(x, y) E(Q) : x = a, h(a), h(b) < B} b grows like C log(b) rank(e)/2. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

24 Rank 28 (Elkies) Currently the largest known rank is (at least) 28: E : y 2 = x 3 + ax + b a = b = P 1 = ( , ) P 2 = ( , ) P 3 = ( , ) P 4 = ( , ) P 5 = ( , ) P 6 = ( , ) P 7 = ( , ) P 8 = ( , ) P 9 = ( , ) P 10 = ( , ) P 11 = ( , ) P 12 = ( , ) P 13 = ( , ) P 14 = ( , ) P 15 = ( , ) Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

25 Counting points modulo p Instead of trying to count E(Q), for primes p count E(Z p ) := {(x, y) Z p Z p : y 2 x 3 + ax + b (mod p)} { } Example E : y 2 = x 3 + 2x + 1 p = 5 Zso 5 ) E(Z 5 ) has 7 points. x x 3 + 2x + 1 (mod 5) y 0 1 1, , , Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

26 Counting points modulo p Theorem (Gauss) If E d is the elliptic curve y 2 = x 3 d 2 x and p 2d, then #(E(Z p )) = p + 1 if p 3 (mod 4), #(E(Z p )) = p ( d p) u if p 1 (mod 4), where ) is the Legendre symbol, ( d p p = u 2 + v 2, v is even, and u v + 1 (mod 4). Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

27 Idea of Birch and Swinnerton-Dyer There is a reduction map E(Q) E(Z p ). Birch and Swinnerton-Dyer suggested that the larger E(Q) is, the larger the E(Z p ) should be on average. How can we measure this? Birch and Swinnerton-Dyer computed as X grows. p<x #(E(Z p )) p Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

28 Data for y 2 = x 3 d 2 x 20 d = 6 d = 5 d = 7 10 #(E d (Z p)) p<x p X d = 157 d = 1 d = 2 d = 3 Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

29 Better idea Define the L-function of E L(E, s) := ( p #(E(Z p)) + p ) 1. p s p 2s p As a function of the complex variable s, this product converges on the half-plane Re(s) > 3/2. If we set s = 1 (!!!) L(E, 1) = p ( #(E(Zp )) p ) 1. This is (the inverse of) what Birch and Swinnerton-Dyer were computing. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

30 Better idea The Birch and Swinnerton-Dyer heuristic predicts that L(E, 1) should tell us how big E(Q) is. Theorem (Wiles et al., 1999) L(E, s) has an analytic continuation to the entire complex plane. Conjecture (Birch and Swinnerton-Dyer) E(Q) is infinite if and only if L(E, 1) = 0. (In fact, they conjecture that rank(e) = ord s=1 L(E, s).) Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

31 Theorem Theorem (Coates & Wiles, Kolyvagin, Kato,... ) If E(Q) is infinite, then L(E, 1) = 0. Now let E d be the elliptic curve y 2 = x 3 d 2 x, where d is a positive squarefree integer. Corollary If there is a right triangle with rational sides and area d, then L(E d, 1) = 0. We need a way to evaluate L(E d, 1). Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

32 L(E d, 1) Theorem (Tunnell) Let E d be the elliptic curve y 2 = x 3 d 2 x. Then L(E d, 1) = a(n 2m)2 16 d 1 dx x3 x where a = 1 if d is odd, and a = 2 if d is even, n = #{(x, y, z) Z 3 : x 2 + 2ay 2 + 8z 2 = d/a}, m = #{(x, y, z) Z 3 : x 2 + 2ay z 2 = d/a}. Corollary If there is a right triangle with rational sides and area d, then n = 2m. Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

33 Open questions 1 Prove the converse: if n = 2m, then there is a rational right triangle with area d. 2 If n = 2m, find a rational right triangle with area d. (There is a method that works most (?) of the time, including d = 157, but not always.) 3 How often is there a rational right triangle with area d, if d 1, 2, or 3 (mod 8)? (Guess: the number of such d < X is about X 3/4 ). Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

34 Right triangles and elliptic curves Karl Rubin Ross Reunion July 2007 Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July / 34

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