On Arithmetic Progressions on Genus Two Curves

We study arithmetic progression in the x-coordinate of rational points on genus two curves. As we know, there are two models for the curve C of genus two: C : y = f5(x) or C : y = f6(x), where f5, f6 ∈ Q[x], deg f5 = 5, deg f6 = 6 and the polynomials f5, f6 do not have multiple roots. First we prove that there exists an infinite family of curves of the form y = f(x), where f ∈ Q[x] and deg f = 5 each containing 11 points in arithmetic progression. We also present an example of F ∈ Q[x] with deg F = 5 such that on the curve y = F (x) twelve points lie in arithmetic progression. Next, we show that there exist infinitely many curves of the form y = g(x) where g ∈ Q[x] and deg g = 6, each containing 16 points in arithmetic progression. Moreover, we present two examples of curves in this form with 18 points in arithmetic progression.