Twists and Generalized Zolotarev Polynomials

In the previous article [3], the author investigated Diophantine equations of the form (∗) X − f(x)Y 2 = 1 and found that the set of the polynomial solutions of (∗) can be described in terms of the theory of twists and that of generalized Jacobian varieties. On the other hand, there are several articles ([1], [7]) which relate the solutions of (∗) with certain torsion points on the Jacobian variety of the hyperelliptic curve y2 = f(x). One of the purposes of the present paper is to show that our viewpoint reveals quite naturally the reason why the torsion points show up in this connection. Another purpose is to generalize the notion of the socalled Zolotarev polynomials and investigate the extremal property, which is studied in [11], and the arithmetic properties of the generalized ones. The original polynomials are investigated in [4] as the set of solutions of (∗) when deg f = 4, in connection with the universal family the elliptic curves y2 = f(x) with a specified torsion point, hence with the arithmetic of the modular curve X1(N). Furthermore, it is pointed out in [10] that, when the base field is the field of complex numbers, (∗) is related with certain planar graphs through the notion of “dessins d’enfants”. In the present paper, generalizing the method in [3], we will see the theory of twists provides us with natural and unified viewpoint to investigate these types of problems for polynomials f(x) of any degree. The plan of this paper is as follows. In Section 2 we recall the definition of twists and their fundamental properties. In Section 3, we will explain the reason why the polynomial solutions of (∗) are related with torsion divisors on the Jacobian variety of the hyperelliptic curve y2 = f(x), in terms of the theory of twists and that of the generalized Jacobian varieties. Section 4 is devoted to the investigation of extremal properties of the solutions. It