Topics in the theory of p-adic heights on elliptic curves

Abstract

This thesis deals with several theoretical and computational problems in the theory of p-adic heights on elliptic curves. In Chapter 3 we provide a new algorithm for the computation of the Mazur--Tate p-adic height in a family of elliptic curves over the rationals. As an application, we study the conjectural non-degeneracy properties of the pairing induced by the height, as well as orders of vanishing and arithmetic interpretations of some of the coefficients of the regulator series. In Chapter 4 we use the decomposition of the p-adic height as a sum of local contributions to explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. For instance, we remove the assumption of semistability in the description of the quadratic Chabauty sets X(Z_p)₂ containing the integral points X(Z) of an elliptic curve of rank at most 1. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference X(Z_p)₂\X(Z) In Chapter 5 we describe an implementation of the Chabauty--Coleman method for genus 3 hyperelliptic curves over Q whose Jacobians have rank 1. This is used to compute the rational points on approximately 17,000 curves and to provide evidence for a conjecture of Stoll. In Chapter 6 we extend the explicit quadratic Chabauty techniques to compute integral points on elliptic curves and rational points on genus 2 bielliptic curves to general number fields. Finally, in Chapter 7 we prove that the first two coefficients in the series expansion around s=1 of the p-adic L-function of an elliptic curve over the rationals are related by a formula involving the conductor of the curve. The introductory Chapter 1 outlines the common themes of the thesis; Chapter 2 introduces some preliminary notions.