The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s, E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a ...

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 ran...

In David Hilbert’s lecture in 1900, he presented 15 difficult problems that guided much of the research of mathematicians in the last century. Today, only 12 of these 15 problems have been solved. In honor of this great mathematician and to celebrate mathematics of the new millennium, the Clay Mathematics Institute of Cambridge, Massachusetts is offering a one million dollar prize for the solut...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e., such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mord...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank zero, i.e., such that the L-function LE(s) of E does not vanish at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to E modulo a power r of a prime p. The theory of vis...

Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1...