Integer points on curves of genus two and their Jacobians

Grant, David

doi:10.1090/S0002-9947-1994-1184116-2

Integer points on curves of genus two and their Jacobians
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Trans. Amer. Math. Soc. 344 (1994), 79-100 Request permission

Abstract:

Let C be a curve of genus 2 defined over a number field, and $\Theta$ the image of C embedded into its Jacobian J. We show that the heights of points of J which are integral with respect to ${[2]_\ast }\Theta$ can be effectively bounded. As a result, if P is a point on C, and $\bar P$ its image under the hyperelliptic involution, then the heights of points on C which are integral with respect to P and $\bar P$ can be effectively bounded, in such a way that we can isolate the dependence on P, and show that if the height of P is bigger than some bound, then there are no points which are S-integral with respect to P and $\bar P$. We relate points on C which are integral with respect to P to points on J which are integral with respect to $\Theta$, and discuss approaches toward bounding the heights of the latter.

References

A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc. 65 (1969), 439–444. MR 234912, DOI 10.1017/s0305004100044418

An introduction to the theory of multiply periodic functions

J. Coates, Construction of rational functions on a curve, Proc. Cambridge Philos. Soc. 68 (1970), 105–123. MR 258831, DOI 10.1017/s0305004100001110
Kevin R. Coombes and David R. Grant, On heterogeneous spaces, J. London Math. Soc. (2) 40 (1989), no. 3, 385–397. MR 1053609, DOI 10.1112/jlms/s2-40.3.385
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353, DOI 10.2307/2944319
Eugene Victor Flynn, The Jacobian and formal group of a curve of genus $2$ over an arbitrary ground field, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 3, 425–441. MR 1041476, DOI 10.1017/S0305004100068729
E. V. Flynn, The group law on the Jacobian of a curve of genus $2$, J. Reine Angew. Math. 439 (1993), 45–69. MR 1219694, DOI 10.1515/crll.1993.439.45
William Fulton, Algebraic curves, Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. An introduction to algebraic geometry; Notes written with the collaboration of Richard Weiss; Reprint of 1969 original. MR 1042981
Daniel M. Gordon and David Grant, Computing the Mordell-Weil rank of Jacobians of curves of genus two, Trans. Amer. Math. Soc. 337 (1993), no. 2, 807–824. MR 1094558, DOI 10.1090/S0002-9947-1993-1094558-0
David Grant, Formal groups in genus two, J. Reine Angew. Math. 411 (1990), 96–121. MR 1072975, DOI 10.1515/crll.1990.411.96
K. Győry, On the number of solutions of linear equations in units of an algebraic number field, Comment. Math. Helv. 54 (1979), no. 4, 583–600. MR 552678, DOI 10.1007/BF02566294
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0

Kummer’s quartic surface

Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2

Algebraic number theory

David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651, DOI 10.1007/978-1-4899-2843-6
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
Wolfgang M. Schmidt, Construction and estimation of bases in function fields, J. Number Theory 39 (1991), no. 2, 181–224. MR 1129568, DOI 10.1016/0022-314X(91)90044-C
Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR 1176315, DOI 10.1007/BFb0098246
Wolfgang M. Schmidt, Integer points on curves of genus $1$, Compositio Math. 81 (1992), no. 1, 33–59. MR 1145607
Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. MR 1002324, DOI 10.1007/978-3-663-14060-3
Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722

Über einige Anwendungen diophantischer Approximationen

The integer solutions of the equation

1

Joseph H. Silverman, Integral points on abelian varieties, Invent. Math. 81 (1985), no. 2, 341–346. MR 799270, DOI 10.1007/BF01389056
Joseph H. Silverman, Integral points on abelian surfaces are widely spaced, Compositio Math. 61 (1987), no. 2, 253–266. MR 882977
Joseph H. Silverman, Integral points on curves and surfaces, Number theory (Ulm, 1987) Lecture Notes in Math., vol. 1380, Springer, New York, 1989, pp. 202–241. MR 1009803, DOI 10.1007/BFb0086555
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
Paul M. Voutier, An upper bound for the size of integral solutions to $Y^m=f(X)$, J. Number Theory 53 (1995), no. 2, 247–271. MR 1348763, DOI 10.1006/jnth.1995.1090