On a theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over genus fields of real quadratic fields

In this paper, we remove certain hypotheses 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.