The Birch and Swinnerton-Dyer conjecture famously predicts that the rank of an elliptic curve, or more generally an abelian variety, can be computed from its L- function. A consequence of this, known as the parity conjecture, is a purely arithmetic result which bypasses the conjectural theory of L-functions and asserts that the parity of the rank is determined by the root number.

This thesis investigates the parity conjecture for Jacobians of hyperelliptic curves and collates some of the first pieces of evidence (beyond elliptic curves) for the Birch and Swinnerton-Dyer conjecture. In doing this, we exhibit formulae for the parity of the rank of certain abelian varieties which use only the local theory of curves.