ON HIGH RANK pi/3 AND 2 pi/3-CONGRUENT NUMBER ELLIPTIC CURVES

摘要

Consider the elliptic curves given by E-n,E-theta : y(2) = x(3) + 2snx(2) - (r(2) - s(2))n(2)x where 0 < theta < pi, COS(theta) = s/r is rational with 0 <= |s| < r and gcd(r, s) = 1. These elliptic curves are related to the theta-congruent number problem as a generalization of the congruent number problem. For fixed theta, this family corresponds to the quadratic twist by n of the curve E-theta : y(2) = x(3) + 2sx(2) - (r(2) - s(2))x. We study two special cases: theta = pi/3 and theta = 2 pi/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E-n,E-theta has rank up to 7 over Q in both cases.