This talk will discuss developments in the theory of local
arithmetic constants associated to an elliptic curve E over a number field
k, as introduced and studied by Mazur and Rubin. I calculate the
arithmetic constant for places of k where E has bad reduction, giving a
more general setting in which one has a lower bound for the rank of the
p-power Selmer group of E over extensions of k. Also, by comparing the
local arithmetic constants with the local analytic root numbers of E, I
determine a setting in which one can verify a (relative) parity conjecture
for E.