Minimal heights and regulators for elliptic surfaces

For each of n = 1, 2, 3 we find the minimal height h ( P ) of a non-torsion point of an elliptic curve E over [Special characters omitted.] ( t ) of discriminant degree 12 n (equivalently of arithmetic genus n ) with 2- or 3-torsion point T. In each case we exhibit the ( E, P, T ) attaining the minimum. For n = 1, 2 we determine the minimal regulator for a rank 2 subgroup [Special characters omitted.] of an elliptic curve E over [Special characters omitted.] ( t ) of discriminant degree 12 n. In each case we exhibit the ( E, P, Q ) that attains the minimum. In both the torsion and rank 2 cases, we also prove that these are the minima for an elliptic curve of discriminant degree 12 n over a function field [Special characters omitted.] ( C ) of any genus. In the torsion setting, the optimal curves are characterized by having the first several multiples of their non-torsion sections being integral. In the rank 2 setting the optimal curves have the greatest number of integral combinations mP + m'Q. In the asymptotic setting we find some new values for the constant C K appearing in the conjecture of Lang which postulates a uniform lower bound for the canonical height of non-torsion points on elliptic curves. We determine values of C K in the case that E is an elliptic curve over K = k ( C ) with a 2- or 3-torsion point, and conjecture the best possible values. In addition, we make progress in the rank 2 case, outlining a strategy by which one can surmise and hopefully prove a conjecture about the possible Mordell-Weil lattices, i.e. the asymptotically obtainable region in the 3-dimensional space of reduced 2-dimensional quadratic forms.