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

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]**D. Abramovich and J. Harris,*Abelian varieties and curves in*, Comp. Math.**78**(1991:2), 227-238. MR**92c:14022****[A-C-G-H]**E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris,*Geometry of algebraic curves I*, Springer-Verlag, New York, 1985. MR**86h:14019****[Ar]**M. Artin,*Lipman's proof of resolution of singularities for surfaces*, in*Arithmetic geometry*(edited by G. Cornell and J. Silverman), Springer-Verlag, New York, 1986, 267-288. MR**89b:14029****[Ch]**T. Chinburg,*An introduction to Arakelov intersection theory*, in*Arithmetic geometry*(edited by G. Cornell and J. Silverman), Springer-Verlag, New York, 1986, 289-308. MR**89b:14029****[D-F]**O. Debarre and R. Fahlaoui,*Abelian varieties in**and points of bounded degree on algebraic curves,*Comp. Math.,**88**(1993), 235-249. MR**94h:14028****[Fa 1]**G. Faltings,*Diophantine approximation on abelian varieties*, Ann. of Math.**133**(1991:2), 549-576. MR**93d:11066****[Fa 2]**G. Faltings,*The general case of S. Lang's conjecture*, in Christante, V. and Messing, W. (eds.),*Barsotti symposium in algebraic geometry*, Perspectives in Mathematics**15**, Academic Press, San Diego, Calif., 1994, 175-182. MR**95m:11061****[Frey]**G. Frey,*Curves with infinitely many points of fixed degree*, Israel J. Math.,**85**(1994), 79-83. MR**94m:11072****[Fu]**W. Fulton,*Intersection Theory*, Springer-Verlag, Berlin, 1984. MR**85k:14004****[H-S]**J. Harris and J. Silverman,*Bielliptic curves and symmetric products*, Proc. Amer. Math. Soc.,**112**(1991:2), 347-356. MR**91i:11067****[Ha]**R. Hartshorne,*Algebraic geometry*, Springer-Verlag, Graduate Texts in Mathematics, vol. 52, New York, 1977. MR**57:3116****[Kani]**E. Kani,*The number of curves of genus 2 with elliptic differentials*, J. Reine Angew. Math.,**485**(1997), 93-121. MR**98g:14025****[L 1]**S. Lang,*Fundamentals of diophantine geometry*, Springer-Verlag, New York, 1983. MR**85j:11005****[L 2]**S. Lang,*Introduction to Arakelov theory*, Springer-Verlag, New York, 1988. MR**89m:11059****[Ma]**G. Martens,*On coverings of elliptic curves*, in*Algebra and number theory (Essen, 1992)*, de Gruyter, Berlin, 1994, 137-151. MR**95e:14024****[Si]**J. Silverman,*Rational points on symmetric products of a curve*, Amer. J. Math.**113**(1991), 471-508. MR**92m:11060****[S-T]**X. Song and T. J. Tucker,*Dirichlet's Theorem, Vojta's inequality, and Vojta's conjecture*, Comp. Math.**116**(1999:2), 219-238. MR**2000d:11085****[Tu]**T. J. Tucker,*Generalizations of Hilbert's irreducibility theorem*, preprint.**[V 1]**P. Vojta,*Diophantine approximations and value distribution theory*, Lecture Notes in Math., vol. 1239, Springer-Verlag, New York, 1987. MR**91k:11049****[V 2]**P. Vojta,*Arithmetic discriminants and quadratic points on curves*, in*Arithmetic algebraic geometry*(Texel, 1989), Progr. Math.**89**, Birkhäuser Boston, Boston, MA, 1991, 359-376. MR**92j:11059****[V 3]**P. Vojta,*A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing*, J. Amer. Math. Soc.,**5**(1992:4), 763-804. MR**94a:11093****[Zh]**S. Zhang, Note to G. Frey, 1994.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11G30,
11J25

Retrieve articles in all journals with MSC (2000): 11G30, 11J25

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