Models of curves : the birch and Swinnerton-Dyer conjecture & ordinary reduction

Abstract

Chapter 1,contains the numerical verification of the Birch

and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic

curves of genus 2, 3, 4 and 5. Chapter 2 treats the equivalence of BSD for a

certain elliptic curve over Q(∜5), and a pair of hyperelliptic curves over Q. In chapter 3, ordinary Galois covers of smooth curves are constructed from ordinary Galois covers of semi-stable curves. Finally, in chapter 4, the statistics of ordinary reduction for hyperelliptic curves is considered.