The BSD Conjecture

William Stein

Berkeley January 12, 2000 An elliptic curve “X_{0}(27)”:

y^{2} + y = x^{3} − 7

Cubic curve: Random line intersects at 3 points, including ∞!

Points: ∞, (3, 4), (3, −5) Searched with computer; all I found.

More points over finite fields:

• F_{2} = {0, 1}, y^{2} + y = x^{3} + 1:

∞, (1, 0), (1, 1)

• F_{3} = {0, 1, 2}, y^{2} + y = x^{3} + 2:

∞, (0,1) , (1, 0), (1, 2)

Group law: The points form an abelian group. No nontrivial map Z/3Z → Z/4Z!! F = y^{2} + y − x^{3} − 2 = 0, F_{x} = 0 = F_{y} = 2y + 1 = 0 at singular point (0, 1). Tamagawa number c_{3} = 3, measures type of reduction.

• F_{5}, y^{2} + y = x^{3} + 3

{∞, (2, 2), (3, 0), (3, 4), (4, 1), (4, 3)}

Cryptography: Schaefer’s course, Koblitz, SEA, discrete log problem, Lenstra’s ECM.