Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication
The authors show that for each odd prime \(l\), each \(\varepsilon>0\) and for all sufficiently large \(X>0\), the number of imaginary quadratic fields of discriminant less than~\(X\) and class number \textit{not} divisible by \(l\) is at least \(\left({{2(l-2)}\over{\sqrt{3}(l-1)}}-\varepsilon\right){{\sqrt{X}}\over {\text{ log} X}}\). They also show that for each elliptic curve \(E\) with complex multiplication, all sufficiently large primes~\(l\) and all sufficiently large \(X>0\), the number of imaginary quadratic fields \(k\) of discriminant less than \(X\) for which the twisted elliptic curve \(E_{(k)}\) has finite Mordell-Weil group \(E_{(k)}({\mathbb Q})\) and has Tate-Shafarevich group of order prime to~\(l\), is at least \(C_{\varepsilon,l}{{\sqrt{X}}\over{\log X}}\) for some constant \(C_{\varepsilon,l}\) that only depends on \(\varepsilon\) and \(l\). The results follow by comparing coefficients of two modular forms of weight \(3/2\) and applying a theorem of J.~Sturm's concerning modular forms modulo~\(p\).
scientific article; zbMATH DE number 1275764