  Poonen and Stoll have shown that the reduced Shafarevich-Tate group of a
principally polarized abelian variety over a global field can have order twice
a square (the odd case) as well as a square (the even case). For a curve over a
global field, they give a local diophantine criterion for its jacobian to be
even or odd. In this note we use the Cherednik-Drinfeld p-adic uniformization
to study the local points and divisors on certain Atkin-Lehner quotients of
Shimura curves. We exhibit an explicit family of such quotient curves over the
rationals with odd jacobians, with genus going to infinity, and with at most
finitely many hyperelliptic members.
