Non-trivial zeroes for quadratic twists of Hasse-Weil \(L\)-functions
Let \(E\) be an elliptic curve defined over \({\mathbb Q}\), endowed with a model \(y^2=x^3+Ax+B\) with \(A,B\in{\mathbb Q}\). This article deals with the problem of the distribution of numbers \(D\) such that the rank of the Mordell-Weil group of the \(D\)-th quadratic twist \(E_D\) is positive. Here sieve techniques lead to lower bounds for the number of fundamental discriminants with prescribed number of prime divisors and bounded absolute value, so that the Hasse-Weil \(L\)-function of \(E_D\) vanishes at \(s=1\).
rank of elliptic curves over \({\mathbb Q}\)
scientific article; zbMATH DE number 1375114