Rank Zero Quadratic Twists of Modular Elliptic Curves

In [11] L. Mai and M. R. Murty proved that if E is a modular elliptic curve with conductor N, then there exists infinitely many square-free integers D ≡ 1 mod 4N such that ED, the D−quadratic twist of E, has rank 0. Moreover assuming the Birch and Swinnerton-Dyer Conjecture, they obtain analytic estimates on the lower bounds for the orders of their Tate-Shafarevich groups. However regarding ranks, simply by the sign of functional equations, it is not expected that there will be infinitely many square-free D in every arithmetic progression r (mod t) where gcd(r, t) is square-free such that ED has rank zero. Given a square-free positive integer r, under mild conditions we show that there exists an integer tr and a positive integer N where tr ≡ r mod Q× 2 p for all p | N, such that there are infinitely many positive square-free integers D ≡ tr mod N ′ where ED has rank zero. The modulus N ′ is defined by N ′ := 8 Q p|N p where the product is over odd primes p dividing N. As an example of this theorem, let E be defined by