An example of elliptic curve over $Q$ with rank $ \geqq 20$

An example of elliptic curve over Q with rank equal to 15

We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15.

THE RANK AND GENERATORS OF KIHARA’S ELLIPTIC CURVE WITH TORSION Z/4Z OVER Q(t)

For the elliptic curve E overQ(t) found by Kihara, with torsion group Z/4Z and rank ≥ 5, which is the current record for the rank of such curves, by using a suitable injective specialization, we determine exactly the rank and generators of E(Q(t)).

K 3 surfaces with Picard rank 20 over Q

We compute all K3 surfaces with Picard rank 20 over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with geometric Picard rank 20, but not necessaril...

Elliptic Curves over Q

All polynomials and rational functions in this essay are assumed to have coefficients in Q . Fix an integer n ≥ 1. An affine variety is a simultaneous irreducible system of polynomial equations in n variables. The Q -points, R -points and C points of the affine variety are all solutions of the polynomial system in Q , R n and C , respectively. Rational projective n-spaceg Q n is the set of line...

Elliptic Curves over Q