Heegner points on modular curves (Q2763789)

Let \({\mathfrak O}\) be an order of a quadratic field of discriminant \(D\) and fix an integer \(N\geq 1\). We denote by \(\text{Pic}^+({\mathfrak O})\) the abelian group of invertible fractional \({\mathfrak O}\)-ideals modulo the principal ones defined by generators of positive norm. In this paper the authors define a Heegner triplet of type \((N,D)\) as a triplet \(({\mathfrak O},{\mathfrak n},[{\mathfrak a}])\), where \({\mathfrak n}\) is a primitive \({\mathfrak O}\)-ideal of norm \(N\) and \([{\mathfrak a}]\) an element of \(\text{Pic}^+({\mathfrak O})\), and prove that this concept generalizes the concept of Heegner point on the modular curve \(X_0(N)\). Let \({\mathfrak H}(N,D)\) be the set of integral quadratic forms of type \(aNX^2+ bXY+ cY^2\), with \(D= b^2-4Nac\), \(\gcd(aN,b,c)= \gcd(a,b,Nc)=1\). Furthermore, the authors prove that the group NEWLINE\[NEWLINE\Gamma_0(N)= \left\{ \left[ \begin{smallmatrix} \alpha&\beta\\ \gamma&\delta \end{smallmatrix} \right]\in \text{SL}(2,\mathbb{Z})\mid \gamma\equiv 0\pmod N\right\}NEWLINE\]NEWLINE operates on \({\mathfrak H}(N,D)\), the quotient \(H(N,D)= {\mathfrak H}(N,D)/ \Gamma_0(N)\) is a finite set and there is a bijection between \(H(N,D)\) and the set of pairs \(({\mathfrak n},[{\mathfrak a}])\). Moreover, when \(D<0\), they obtain a formula for the number of elements of \(H(N,D)\) in terms of the number of representations by genera of binary quadratic forms of discriminant \(D\).