USING ELLIPTIC CURVES TO PRODUCE QUADRATIC NUMBER FIELDS OF HIGH THREE-RANK

We use a connection between the arithmetic of elliptic curves of the form y² = x³ + k and the arithmetic of the quadratic number fields Q(√k) and Q(√-3k) to look for quadratic fields with high three-rank. We give a geometric proof of known results on polynomials that give rise to infinite families of quadratic number fields possessing non-trivial lower bounds on their three-rank. We then generalize the method to produce infinitely many such polynomials. Finally, we produce specific examples of quadratic number fields with high three-rank.