Ranks of Elliptic Curves

William Stein

(joint with Baur Bektemirov, Barry Mazur, and Mark Watkins) December 1, 2004, Harvard NTS

Part I: Arithmetic of Elliptic Curves

2

The Pythagorean Theorem

a^{2} + b^{2} = c^{2} c  a b  _{Pythagoras}_{Approx 569–475BC}

Babylonians

1800-1600BC

Pythagorean Triples

Triples of integers a, b, c such that a_{2} + b_{2} = c_{2}^{(3, 4, 5) ; (5, 12, 13) ; (7, 24, 25) ; (9, 40, 41) ; (11, 60, 61) ; (13, 84, 85) ; (15, 8, 17) ; (21, 20, 29) ; (33, 56, 65) ; (35, 12, 37) ; (39, 80, 89) ; (45, 28, 53) ; (55, 48, 73) ; (63, 16, 65) ; (65, 72, 97) ; (77, 36, 85) ; ⋮}

Enumerating Pythagorean Triples

(−1, 0)  (0, t)(x, y)  \begin{aligned}Slope = t & = \frac{y}{x + 1} \\ x & = \frac{1 − t^{2}}{1 + t^{2}} \\ y & = \frac{2t}{1 + t^{2}} \end{aligned} If t = \frac{r}{s}, then  a = s^{2} − r^{2} b = 2rs c = s^{2} + r^{2} is a Pythagorean triple.