Arithmetic discriminants and morphisms of curves
Authors:
Xiangjun Song and Thomas J. Tucker
Journal:
Trans. Amer. Math. Soc. 353 (2001), 19211936
MSC (2000):
Primary 11G30, 11J25
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 pullbacks 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 bielliptic curves of genus 2.
 [AH]
Dan
Abramovich and Joe
Harris, Abelian varieties and curves in
𝑊_{𝑑}(𝐶), Compositio Math. 78
(1991), no. 2, 227–238. MR 1104789
(92c:14022)
 [ACGH]
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, SpringerVerlag, New York, 1985. MR 770932
(86h:14019)
 [Ar]
Gary
Cornell and Joseph
H. Silverman (eds.), Arithmetic geometry, SpringerVerlag, New
York, 1986. Papers from the conference held at the University of
Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR 861969
(89b:14029)
 [Ch]
Gary
Cornell and Joseph
H. Silverman (eds.), Arithmetic geometry, SpringerVerlag, New
York, 1986. Papers from the conference held at the University of
Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR 861969
(89b:14029)
 [DF]
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
(94h:14028)
 [Fa 1]
Gerd
Faltings, Diophantine approximation on abelian varieties, Ann.
of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353
(93d:11066), http://dx.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
(95m:11061)
 [Frey]
Gerhard
Frey, Curves with infinitely many points of fixed degree,
Israel J. Math. 85 (1994), no. 13, 79–83. MR 1264340
(94m:11072), http://dx.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, SpringerVerlag, Berlin, 1984. MR 732620
(85k:14004)
 [HS]
Joe
Harris and Joe
Silverman, Bielliptic curves and symmetric
products, Proc. Amer. Math. Soc.
112 (1991), no. 2,
347–356. MR 1055774
(91i:11067), http://dx.doi.org/10.1090/S00029939199110557740
 [Ha]
Robin
Hartshorne, Algebraic geometry, SpringerVerlag, New
YorkHeidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
(57 #3116)
 [Kani]
Ernst
Kani, The number of curves of genus two with elliptic
differentials, J. Reine Angew. Math. 485 (1997),
93–121. MR
1442190 (98g:14025), http://dx.doi.org/10.1515/crll.1997.485.93
 [L 1]
Serge
Lang, Fundamentals of Diophantine geometry, SpringerVerlag,
New York, 1983. MR 715605
(85j:11005)
 [L 2]
Serge
Lang, Introduction to Arakelov theory, SpringerVerlag, New
York, 1988. MR
969124 (89m:11059)
 [Ma]
G.
Martens, On coverings of elliptic curves, Algebra and number
theory (Essen, 1992) de Gruyter, Berlin, 1994, pp. 137–151. MR 1285364
(95e:14024)
 [Si]
Joseph
H. Silverman, Rational points on symmetric products of a
curve, Amer. J. Math. 113 (1991), no. 3,
471–508. MR 1109348
(92m:11060), http://dx.doi.org/10.2307/2374836
 [ST]
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
(2000d:11085), http://dx.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, SpringerVerlag,
Berlin, 1987. MR
883451 (91k:11049)
 [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
(92j:11059), http://dx.doi.org/10.1007/PL00011403
 [V 3]
Paul
Vojta, A generalization of theorems of
Faltings and ThueSiegelRothWirsing, J. Amer.
Math. Soc. 5 (1992), no. 4, 763–804. MR 1151542
(94a:11093), http://dx.doi.org/10.1090/S08940347199211515429
 [Zh]
S. Zhang, Note to G. Frey, 1994.
 [AH]
 D. Abramovich and J. Harris, Abelian varieties and curves in , Comp. Math. 78 (1991:2), 227238. MR 92c:14022
 [ACGH]
 E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, Geometry of algebraic curves I, SpringerVerlag, 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), SpringerVerlag, New York, 1986, 267288. MR 89b:14029
 [Ch]
 T. Chinburg, An introduction to Arakelov intersection theory, in Arithmetic geometry (edited by G. Cornell and J. Silverman), SpringerVerlag, New York, 1986, 289308. MR 89b:14029
 [DF]
 O. Debarre and R. Fahlaoui, Abelian varieties in and points of bounded degree on algebraic curves, Comp. Math., 88 (1993), 235249. MR 94h:14028
 [Fa 1]
 G. Faltings, Diophantine approximation on abelian varieties, Ann. of Math. 133 (1991:2), 549576. 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, 175182. MR 95m:11061
 [Frey]
 G. Frey, Curves with infinitely many points of fixed degree, Israel J. Math., 85 (1994), 7983. MR 94m:11072
 [Fu]
 W. Fulton, Intersection Theory, SpringerVerlag, Berlin, 1984. MR 85k:14004
 [HS]
 J. Harris and J. Silverman, Bielliptic curves and symmetric products, Proc. Amer. Math. Soc., 112 (1991:2), 347356. MR 91i:11067
 [Ha]
 R. Hartshorne, Algebraic geometry, SpringerVerlag, 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), 93121. MR 98g:14025
 [L 1]
 S. Lang, Fundamentals of diophantine geometry, SpringerVerlag, New York, 1983. MR 85j:11005
 [L 2]
 S. Lang, Introduction to Arakelov theory, SpringerVerlag, 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, 137151. MR 95e:14024
 [Si]
 J. Silverman, Rational points on symmetric products of a curve, Amer. J. Math. 113 (1991), 471508. MR 92m:11060
 [ST]
 X. Song and T. J. Tucker, Dirichlet's Theorem, Vojta's inequality, and Vojta's conjecture, Comp. Math. 116 (1999:2), 219238. 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, SpringerVerlag, 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, 359376. MR 92j:11059
 [V 3]
 P. Vojta, A generalization of theorems of Faltings and ThueSiegelRothWirsing, J. Amer. Math. Soc., 5 (1992:4), 763804. MR 94a:11093
 [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:
http://dx.doi.org/10.1090/S000299470102709X
PII:
S 00029947(01)02709X
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
