  In the early 1980s, Rohrlich began a study of canonical Hecke characters,
which are closely related to the simplest examples of CM elliptic curves. He
and Montgomery showed the non-vanishing of the central value when the
L-function has an even functional equation, and we now show the non-vanishing
of the central derivative when the functional equation is odd. Using the
results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing
to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by
Gross in his thesis. In particular, their rank is determined by a congruence
condition. http://www.math.yale.edu/users/steve/milleryang
