Algebra & Number Theory

Tubular approaches to Baker's method for curves and varieties

Samuel Le Fourn

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we generalize results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in p-adic logarithms. We then use these ideas to improve known estimates on solutions of S-unit equations. Finally, we explain how a finer analysis and formalism can improve upon the conditions given, and give some applications to the Siegel modular variety A2(2).

Article information

Source
Algebra Number Theory, Volume 14, Number 3 (2020), 763-785.

Dates
Received: 18 February 2019
Revised: 20 August 2019
Accepted: 7 October 2019
First available in Project Euclid: 2 July 2020

Permanent link to this document
https://projecteuclid.org/euclid.ant/1593655271

Digital Object Identifier
doi:10.2140/ant.2020.14.763

Mathematical Reviews number (MathSciNet)
MR4113780

Subjects
Primary: 11G35: Varieties over global fields [See also 14G25]
Secondary: 11J86: Linear forms in logarithms; Baker's method

Keywords
integral points Baker's method Runge's method $S$-unit equation

Citation

Le Fourn, Samuel. Tubular approaches to Baker's method for curves and varieties. Algebra Number Theory 14 (2020), no. 3, 763--785. doi:10.2140/ant.2020.14.763. https://projecteuclid.org/euclid.ant/1593655271


Export citation

References

  • Y. Bilu, “Effective analysis of integral points on algebraic curves”, Israel J. Math. 90:1-3 (1995), 235–252.
  • Y. Bugeaud and K. Győry, “Bounds for the solutions of unit equations”, Acta Arith. 74:1 (1996), 67–80.
  • P. Corvaja, V. Sookdeo, T. J. Tucker, and U. Zannier, “Integral points in two-parameter orbits”, J. Reine Angew. Math. 706 (2015), 19–33.
  • J.-H. Evertse and K. Győry, Unit equations in Diophantine number theory, Cambridge Studies in Advanced Mathematics 146, Cambridge University Press, 2015.
  • G. van der Geer, “On the geometry of a Siegel modular threefold”, Math. Ann. 260:3 (1982), 317–350.
  • K. Győry, “Bounds for the solutions of $S$-unit equations and decomposable form equations II”, preprint, 2019.
  • K. Győry and K. Yu, “Bounds for the solutions of $S$-unit equations and decomposable form equations”, Acta Arith. 123:1 (2006), 9–41.
  • J.-i. Igusa, “On Siegel modular forms genus two, II”, Amer. J. Math. 86 (1964), 392–412.
  • J.-i. Igusa, “On the graded ring of theta-constants”, Amer. J. Math. 86 (1964), 219–246.
  • S. Lang, Fundamentals of Diophantine geometry, Springer, 1983.
  • S. Le Fourn, “A tubular variant of Runge's method in all dimensions, with applications to integral points on Siegel modular varieties”, Algebra Number Theory 13:1 (2019), 159–209.
  • A. Levin, “Variations on a theme of Runge: effective determination of integral points on certain varieties”, J. Théor. Nombres Bordeaux 20:2 (2008), 385–417.
  • A. Levin, “Linear forms in logarithms and integral points on higher-dimensional varieties”, Algebra Number Theory 8:3 (2014), 647–687.
  • A. Levin, “Extending Runge's method for integral points”, pp. 171–188 in Higher genus curves in mathematical physics and arithmetic geometry, edited by A. Malmendier and T. Shaska, Contemp. Math. 703, Amer. Math. Soc., Providence, RI, 2018.
  • Q. Liu, “Courbes stables de genre $2$ et leur schéma de modules”, Math. Ann. 295:2 (1993), 201–222.
  • D. W. Masser and G. Wüstholz, “Fields of large transcendence degree generated by values of elliptic functions”, Invent. Math. 72:3 (1983), 407–464.
  • J. H. Silverman, “Arithmetic distance functions and height functions in Diophantine geometry”, Math. Ann. 279:2 (1987), 193–216.
  • M. Streng, Complex multiplication of abelian surfaces, Ph.D. thesis, Universiteit Leiden, 2010, https://openaccess.leidenuniv.nl/handle/1887/15572.
  • P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics 1239, Springer, 1987.