Integer points on complements of dual curves and on genus one modular curves

Abstract

Higher dimension analogs of Siegel's theorem on the finiteness of integer points are known in limited cases and often come with restrictions on the divisor at infinity, e.g. that the divisor should have many irreducible components. In the first half of this thesis, we give a new class of prime divisors in the plane, namely duals of certain smooth plane curves, whose complements have finitely many integer points. This is accomplished using a moduli of curves interpretation. In the second half of this thesis, we give a proof of the finiteness of integer points on genus one modular curves. This result is not new, but the proof we give is based on the $p$-adic period map and ideas from a recent new proof of the Mordell conjecture by Lawrence and Venkatesh.