Arithmetic Discriminants and Morphisms of Curves

This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pullbacks of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent algebraic points. It is also shown that bounds on the arithmetic discriminant yield information about how the fields of definition k(P ) and k(f(P )) differ when P is an algebraic point on a curve C and f : C −→ C′ is a nonconstant morphism of curves. In particular, it is demonstrated that k(P ) #= k(f(P )), with at most finitely many exceptions, whenever the degrees of P and f are sufficiently small, relative to the difference between the genera g(C) and g(C′). The paper concludes with a detailed analysis of the arithmetic discriminants of quadratic points on bi-elliptic curves of genus 2. Let C be a curve defined over a number field k, and let X be a regular model for C over the ring of integers R of k. In [V 3], Vojta proves that, for any ! > 0, all points P ∈ C(k̄) of bounded degree satisfy the inequality hK(P ) ≤ da(P ) + !hA(P ) + O(1), (0.0.1) where A is any ample divisor on C, hA and hK are Weil heights associated with A and the canonical divisor K, respectively, and da(P ) is the arithmetic discriminant of P . The arithmetic discriminant depends on the choice of a regular model X ; it is defined as (HP .(ωX/B + HP )) [k(P ) : k] , where HP is the arithmetic curve on X corresponding to P and ωX/B is the sheaf of relative differentials of X over B = Spec R. The difference between arithmetic discriminants derived from different regular models is always bounded. Hence, (0.0.1) is not affected by one’s choice of a regular model. The arithmetic discriminant is related to d(P ), the normalized field discriminant of the field of definition of P , which is defined as d(P ) := log |N QDk(P )/k| [k(P ) : Q] . One may obtain da(P ) by adding to d(P ) terms corresponding to singularities of HP at finite places and v-adic distances between the conjugates of P at infinite places v (see [V 2, Section 3]). The precise nature of the relationship between Received by the editors November 30, 1999 and, in revised form, February 25, 2000. 2000 Mathematics Subject Classification. Primary 11G30, 11J25. c ©2001 American Mathematical Society