Elliptic curves are a fundamental object of study in modern number theory. Elliptic curves are parametrized by another class of curves known as modular curves. That is, each point on a modular curve corresponds to an elliptic curve together with additional level structure.
This IGL project will consider some specific examples of modular curves and attempt to understand their rational points, thus giving information on families of elliptic curves. We will attempt to understand modular curves with twisted rational level 2 and level 3 structures, and try to understand when they determine rational elliptic curves with a corresponding level 6 structure.
Completion of Math 347. Math 417 or Math 453 would be beneficial, but not necessary.