Parity of ranks of Jacobians of hyperelliptic curves of genus 2

A consequence of the Birch and Swinnerton-Dyer conjecture is that the parity of the rank of abelian varieties is expected to be given by their global root numbers. This is known as the parity conjecture. Assuming the finiteness of the Shafarevich-Tate groups, the parity conjecture is equivalent to the p-parity conjecture for all prime p, that is the p∞ Selmer rank is expected to be given by the global root number. In this thesis we study the parity of the 2∞ Selmer rank of Jacobians of hyperelliptic curves of genus 2 defined over number fields. This forces us to assume the existence of a Richelot isogeny (the analogue of a 2-isogeny for elliptic curves) to provide an expression for the parity of their 2∞ Selmer rank as a sum of local factors Λv modulo 2. Based on a joint work with T. and V. Dokchitser and A. Morgan on arithmetic of hyperelliptic curves over local fields, this makes the parity of the 2∞ Selmer rank of such semistable Jacobians computable in practice. By introducing a set of polynomial invariants in the roots of the defining polynomials of the underlying curves of a specific family of Jacobians, we provide an expression for the local discrepancy existing between the local factors Λv and the local root numbers, and prove the 2-parity conjecture in this case. One outcome of this result it that, using the theory of regulator constants, one can lift the assumption on the existence of an isogeny and prove the parity conjecture for a class of semistable Jacobians of genus 2 curves assuming finiteness of their Shafarevich-Tate group