Mazur Program B

Classify all elliptic curves with Galois groups of some form? Classify l-adic images of Galois.

• Find subgroups $H$that occur as the image of Galois
• Compute equations for modular curves $X_H$
• Determine rational $j$ invariants $j_H: X_H \to X(1) \cong {\mathbf{P}}^1_{/{\mathbf{Q}}}$
• Provably find all rational points on each $X_H$.

Sporadic: not cuspidal and not complex multiplication Compute genus using Riemann-Hurwitz.