Local constants of polarized abelian varieties in dihedral extensions

In the study of the Mordell-Weil group (rational points) of an elliptic curve E, the rank rE of the group remains a rather mysterious object. The Conjecture of Birch and Swinnerton-Dyer (BSD) predicts that rE is equal to the order of the zero at s = 1 of the L-function of E. Studying the modulo 2 version of this conjecture, known as the Parity Conjecture, has been an interesting stepping stone towards understanding the full BSD Conjecture. Following Mazur and Rubin, we study the p-Selmer group of E, rather than the classical Mordell-Weil group of E. The Shafarevich-Tate Conjecture predicts that these groups have the same rank, and so one obtains a parallel p-Selmer Parity Conjecture for the p-Selmer group of E. The work of Rohrlich on local root numbers of the L-function of E, and the work of Mazur and Rubin on arithmetic local constants serve as a basis for our approach to the p-Selmer Parity Conjecture. We also extend the work of Mazur and Rubin which applies the theory of arithmetic local constants to determine lower bounds for the growth of p-Selmer rank in certain extensions of number fields. In another direction, we provide some initial steps toward applying local arithmetic constants to determine lower bounds for the growth of p-Selmer rank of higher dimensional polarized abelian varieties with complex multiplication.