Points on Curves

Abstract

In this thesis we discuss finiteness theorems for integral and rational points on curves over number fields. We do this in two parts.

Part I of the thesis is concerned with upper bounds. We give pointwise upper bounds for the number of: $\o_{K,S}$-points on an elliptic curve (this is joint work with Wei Ho), $\o_{K,S}$-points on a hyperelliptic curve, and large $K$-points on a hyperbolic curve. We also give upper bounds on average for: ranks of elliptic curves in certain thin families, and the number of $\Q$-points on odd genus two curves. For the former we introduce a technique that should allow one to treat counts on invariant quadrics in arithmetic statistics in considerable generality.

Part II of the thesis is concerned with effectivity. We give an algorithm which, on input $(g,K,S)$, outputs the $g$-dimensional abelian varieties over $K$ with good reduction outside $S$. We prove this algorithm always terminates under standard motivic conjectures. (This is joint work with Brian Lawrence.) Using a theorem of Bogomolov-Tschinkel, we give an algorithm that, assuming strong modularity conjectures for $\GL_2$ over all number fields, computes $C(K)$ for $C/K$ a hyperbolic hyperelliptic curve. We give an \emph{unconditional} algorithm that determines the $\o_{K,S}$-points on a Hilbert modular variety in finite time when $K$ is an odd-degree totally real field. Because complete curves in such varieties abound this effectivizes Faltings' Theorem in some cases. Using a construction of Cohen-Wolfart, we give an algorithm that, assuming the existence of motives associated to weight zero cuspidal automorphic representations of $\GL_2$ over CM fields, computes $C(K)$ when $K$ is CM and $C/K$ admits a Belyi map over $K$ with sufficiently divisible ramification degrees. Finally we work out an example in detail: given $K/\Q$ totally real of odd degree and $a\in K^\times$, we explain how to unconditionally compute $C_a(K)$, where $C_a : x^6 + 4y^3 = a^2$.