An a b c d theorem over function fields and applications

Pietro Corvaja; Umberto Zannier

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 4, page 437-454
  • ISSN: 0037-9484

Abstract

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We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u , v , in term of the degree of u , v , which is sharp whenever u , v have few distinct zeros and poles compared to their degree. This sharpens the “ a b c d -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic curves, provided a ≥ 10 4 and c ≥ 2 ; this provides special cases of a known conjecture of Bogomolov.

How to cite

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Corvaja, Pietro, and Zannier, Umberto. "An $abcd$ theorem over function fields and applications." Bulletin de la Société Mathématique de France 139.4 (2011): 437-454. <http://eudml.org/doc/272636>.

@article{Corvaja2011,
abstract = {We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic curves, provided $a\ge 10^4$ and $c\ge 2$; this provides special cases of a known conjecture of Bogomolov.},
author = {Corvaja, Pietro, Zannier, Umberto},
journal = {Bulletin de la Société Mathématique de France},
keywords = {abc conjecture; function fields; curves on algebraic surfaces; S-units},
language = {eng},
number = {4},
pages = {437-454},
publisher = {Société mathématique de France},
title = {An $abcd$ theorem over function fields and applications},
url = {http://eudml.org/doc/272636},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Corvaja, Pietro
AU - Zannier, Umberto
TI - An $abcd$ theorem over function fields and applications
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 4
SP - 437
EP - 454
AB - We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic curves, provided $a\ge 10^4$ and $c\ge 2$; this provides special cases of a known conjecture of Bogomolov.
LA - eng
KW - abc conjecture; function fields; curves on algebraic surfaces; S-units
UR - http://eudml.org/doc/272636
ER -

References

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  1. [1] W. Barth, C. Peters & A. Van de Ven – Compact complex surfaces, Ergebn. Math. Grenzg., vol. 4, Springer, 1984. Zbl0718.14023MR749574
  2. [2] A. Beauville – « Surfaces algébriques complexes », Astérisque 54 (1978). Zbl0394.14014
  3. [3] F. A. Bogomolov – « Holomorphic tensors and vector bundles on projective manifolds », Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), p. 1227–1287, 1439. Zbl0439.14002MR522939
  4. [4] E. Bombieri & W. Gubler – Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge Univ. Press, 2006. Zbl1115.11034MR2216774
  5. [5] W. D. Brownawell & D. W. Masser – « Vanishing sums in function fields », Math. Proc. Cambridge Philos. Soc.100 (1986), p. 427–434. Zbl0612.10010MR857720
  6. [6] P. Corvaja & U. Zannier – « Some cases of Vojta’s conjecture on integral points over function fields », J. Algebraic Geom. 17 (2008), p. 295–333. Addendum in Asian Journal of Math., 14 (2010), p. 581–584. Zbl1221.11146MR2369088
  7. [7] O. Debarre – Higher-dimensional algebraic geometry, Universitext, Springer, 2001. Zbl0978.14001MR1841091
  8. [8] A. Dujella, C. Fuchs & F. Luca – « A polynomial variant of a problem of Diophantus for pure powers », Int. J. Number Theory4 (2008), p. 57–71. Zbl1218.11028MR2387916
  9. [9] U. Zannier – « Some remarks on the S -unit equation in function fields », Acta Arith.64 (1993), p. 87–98. Zbl0786.11019MR1220487

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