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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-01-02709-X
Published electronically: January 4, 2001
MathSciNet review: 1813599
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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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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: https://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

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