  The purpose of the paper is to complete several global and local results
concerning parity of ranks of elliptic curves. Primarily, we show that the
Shafarevich-Tate conjecture implies the parity conjecture for all elliptic
curves over number fields, give a formula for local and global root numbers of
elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in
characteristic 0. The method is to settle the outstanding local formulae by
deforming from local fields to totally real number fields and then using global
parity results.
