  In this paper, we remove certain hypothesis in the theorem of
Bertolini-Darmon on the rationality of Stark-Heegner points over narrow genus
class fields of real quadratic fields. Along the way, we establish that certain
normalized special values of $L$-functions are squares of rational numbers, a
result that is of independent interest, and can be regarded as instances of the
rank zero case of the Birch and Swinnerton-dyer conjecture modulo squares.
