Towers of Curves and Rational Distance Sets

A rational (resp. integral) distance set is a subset S of the plane R such that for all s, t ∈ S, the distance between s and t is a rational number (resp. is an integer). Huff [4] considered rational distance sets S of the following form: given distinct a, b ∈ Q∗, S contains the four points (0,±a) and (0,±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q∗. Such a point (x, 0) must then satisfy the equations x + a = u and x+b = v with u, v ∈ Q. The system of associated homogeneous equations x+az = u and x + bz = v defines a curve C(a, b) of genus 1 in P. Huff, and later his student Peeples [12], provided examples where the elliptic curve C(a, b) has positive rank over Q, thus exhibiting examples of infinite rational distance sets that are not contained in a line or in a circle. These remain to this day the ‘largest’ known such examples. The curves of higher genus whose rational points are related to rational distance sets with 2n + 1 distinct points on the y-axis, (0,±a1), . . . , (0,±an), and (0, 0), plus points (x, 0) on the x-axis, form an interesting class of curves with many rational points and an often computable Mordell-Weil rank over Q. We make some remarks on these curves and on two open problems about rational distance sets. For any field K with char(K) 6= 2, and for α1, . . . , αn ∈ K∗, pairwise distinct, let C(α1, . . . , αn)/K denote the curve in P defined by the system of equations x + αiz 2 = y i , for i = 1, . . . , n. Since char(K) 6= 2 and the coefficients α1, . . . , αn are distinct, the curve C(α1, . . . , αn)/K is smooth. This curve has the following 2 obvious K-rational points