Arithmetic discriminants and morphisms of curves

Authors:
Xiangjun Song and Thomas J. Tucker

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1921-1936

MSC (2000):
Primary 11G30, 11J25

Published electronically:
January 4, 2001

MathSciNet review:
1813599

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 and differ when is an algebraic point on a curve and is a nonconstant morphism of curves. In particular, it is demonstrated that , with at most finitely many exceptions, whenever the degrees of and are sufficiently small, relative to the difference between the genera and . 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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Additional Information

**Xiangjun Song**

Affiliation:
Department of Mathematics, University of California–Berkeley, Berkeley, California 94720

Email:
song@math.berkeley.edu

**Thomas J. Tucker**

Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602

Email:
ttucker@math.uga.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-01-02709-X

Received by editor(s):
November 30, 1999

Received by editor(s) in revised form:
February 25, 2000

Published electronically:
January 4, 2001

Article copyright:
© Copyright 2001
American Mathematical Society