Quadratic forms and elliptic curves, IV

Quadratic Forms and Elliptic Curves

Positive Definite Quadratic Forms, Elliptic Curves and Cubic Congruences

Let F (x, y) = ax + bxy + cy be a positive definite binary quadratic form with discriminant Δ whose base points lie on the line x = −1/m for an integer m ≥ 2, let p be a prime number and let Fp be a finite field. Let EF : y = ax + bx + cx be an elliptic curve over Fp and let CF : ax + bx + cx ≡ 0(mod p) be the cubic congruence corresponding to F . In this work we consider some properties of pos...

Quadratic Twists of Modular Forms and Elliptic Curves

Here we summarize the results presented in the first author’s lecture at the Millenial Conference on Number Theory. These results will be appearing in [O] in full detail. In addition, we present a new result regarding the growth of Tate-Shafarevich groups of certain elliptic curves over elementary abelian simple 2-extensions. We begin by fixing notation. Suppose that F (z) = ∑∞ n=1 a(n)q n ∈ S2...

Modular Forms and Elliptic Curves over Imaginary Quadratic Fields

Elliptic Curves and Modular Forms

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.