Let E: y^2 = x^3 + ax + b be an elliptic curve defined over the rational numbers (so a, b lie in Q). Mordell proved that the group of points E(Q) (which includes the point at infinity) is a finitely generated abelian group, under chord-tangent operations. In 1977, Mazur proved that the torsion subgroup of E(Q) has order at most 16, which allowed him to classify all elliptic curves over Q with a given torsion subgroup (using work of Ogg and Kubert). Mazur's method is to show that the modular curve X_1(p) has no non-cuspidal rational points, for primes p > 7. We will go through a proof of Mazur's theorem. If there is time we will also discuss what one can say about torsion points on higher genus curves (embedded in their Jacobians) as well, the so called Manin-Mumford conjecture (a theorem first proven by Raynaud).
- Teacher: אריאל שנידמן