RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES

Abstract

Let E_m be the family of elliptic curves given by y^2=x^3-x+m^2, which has rank 2 when regarded as an elliptic curve over Q(m). (Here Q represents the field of rational numbers.) Brown and Myers show that a certain quadratic polynomial m(t) has the property that E_m(t) contains an additional rational point that is independent from the two original generators. This implies that there are infinitely many rational numbers n such that E_n(Q) has rank at least 3. We generalize this result, showing that every nonzero rational number n has the property that E_n sits inside such a subfamily of rank 3. Moreover, given any rational point P in E_n, there exists a quadratic polynomial m(t) and a Q(t)-point R(t) in E_m(t) that is independent from the original generators, such that the specialization to t=0 gives m(0)=n and R(0)=P. Such subfamilies can be intersected to increase the rank, demonstrating the existence of a rational subfamily of rank 4 over Q(t), and infinitely many rational numbers n such that E_n(Q) has rank at least 5. Shioda's theory of Mordell-Weil lattices is used to find the generators of such E_m(t) over both Qbar(t) and Q(t) in these cases. (Here Qbar represents the algebraic closure of Q.) All quadratic polynomials m(t) are classified by whether or not E_m(t) contains an additional rational point of low degree. Results similar to these are also obtained for other families of elliptic curves.