Arithmetic discriminants and morphisms of curves

Song, Xiangjun; Tucker, Thomas

doi:10.1090/S0002-9947-01-02709-X

Arithmetic discriminants and morphisms of curves
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by Xiangjun Song and Thomas J. Tucker PDF

Trans. Amer. Math. Soc. 353 (2001), 1921-1936 Request permission

Abstract:

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 pull-backs 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 $f:C \longrightarrow C’$ is a nonconstant morphism of curves. In particular, it is demonstrated that $k(P) \not = k(f(P))$, 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.

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