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

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.

**[A-H]**Dan Abramovich and Joe Harris,*Abelian varieties and curves in 𝑊_{𝑑}(𝐶)*, Compositio Math.**78**(1991), no. 2, 227–238. MR**1104789****[A-C-G-H]**E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,*Geometry of algebraic curves. Vol. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR**770932****[Ar]**Gary Cornell and Joseph H. Silverman (eds.),*Arithmetic geometry*, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR**861969****[Ch]**Gary Cornell and Joseph H. Silverman (eds.),*Arithmetic geometry*, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR**861969****[D-F]**Olivier Debarre and Rachid Fahlaoui,*Abelian varieties in 𝑊^{𝑟}_{𝑑}(𝐶) and points of bounded degree on algebraic curves*, Compositio Math.**88**(1993), no. 3, 235–249. MR**1241949****[Fa 1]**Gerd Faltings,*Diophantine approximation on abelian varieties*, Ann. of Math. (2)**133**(1991), no. 3, 549–576. MR**1109353**, https://doi.org/10.2307/2944319**[Fa 2]**Gerd Faltings,*The general case of S. Lang’s conjecture*, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 175–182. MR**1307396****[Frey]**Gerhard Frey,*Curves with infinitely many points of fixed degree*, Israel J. Math.**85**(1994), no. 1-3, 79–83. MR**1264340**, https://doi.org/10.1007/BF02758637**[Fu]**William Fulton,*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620****[H-S]**Joe Harris and Joe Silverman,*Bielliptic curves and symmetric products*, Proc. Amer. Math. Soc.**112**(1991), no. 2, 347–356. MR**1055774**, https://doi.org/10.1090/S0002-9939-1991-1055774-0**[Ha]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[Kani]**Ernst Kani,*The number of curves of genus two with elliptic differentials*, J. Reine Angew. Math.**485**(1997), 93–121. MR**1442190**, https://doi.org/10.1515/crll.1997.485.93**[L 1]**Serge Lang,*Fundamentals of Diophantine geometry*, Springer-Verlag, New York, 1983. MR**715605****[L 2]**Serge Lang,*Introduction to Arakelov theory*, Springer-Verlag, New York, 1988. MR**969124****[Ma]**G. Martens,*On coverings of elliptic curves*, Algebra and number theory (Essen, 1992) de Gruyter, Berlin, 1994, pp. 137–151. MR**1285364****[Si]**Joseph H. Silverman,*Rational points on symmetric products of a curve*, Amer. J. Math.**113**(1991), no. 3, 471–508. MR**1109348**, https://doi.org/10.2307/2374836**[S-T]**Xiangjun Song and Thomas J. Tucker,*Dirichlet’s theorem, Vojta’s inequality, and Vojta’s conjecture*, Compositio Math.**116**(1999), no. 2, 219–238. MR**1686848**, https://doi.org/10.1023/A:1000948001301**[Tu]**T. J. Tucker,*Generalizations of Hilbert's irreducibility theorem*, preprint.**[V 1]**Paul Vojta,*Diophantine approximations and value distribution theory*, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR**883451****[V 2]**Paul Vojta,*Arithmetic discriminants and quadratic points on curves*, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 359–376. MR**1085268**, https://doi.org/10.1007/PL00011403**[V 3]**Paul Vojta,*A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing*, J. Amer. Math. Soc.**5**(1992), no. 4, 763–804. MR**1151542**, https://doi.org/10.1090/S0894-0347-1992-1151542-9**[Zh]**S. Zhang, Note to G. Frey, 1994.

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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