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

TL;DR

This paper extends Bertolini-Darmon's theorem on the rationality of Stark-Heegner points over real quadratic fields by removing hypotheses and proves that certain special L-values are rational squares, relating to BSD conjecture.

Contribution

It removes hypotheses in a key theorem about Stark-Heegner points and establishes new results on special L-values as rational squares, advancing understanding of rational points and L-values.

Findings

Certain normalized special L-values are squares of rational numbers

Extended theorems on Stark-Heegner points over genus fields

Connected results to the rank zero case of BSD conjecture

Abstract

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.