The Birch–Swinnerton-Dyer Conjecture predicts that, given an abelian variety A over a number field K, its rank, rk(A/K), is equal to the order of vanishing of its L-function L(A/K, s) at s = 1. A consequence of this is the Parity Conjecture; rk(A/K) and the order of vanishing at s=1 of L(A/K, s) are expected to have the same parity. The parity of the latter is given by the root number w(A/K), and so the Parity Conjecture states that (−1)^rk(A/K) = w(A/K).

This thesis investigates what can be said about the Parity Conjecture when A is the Jacobian of a curve of genus 3. Part of this requires developing the local theory of non-hyperelliptic genus 3 curves. We introduce a combinatorial object called an octad diagram, which we conjecture to recover the essential data of stable models.