Superficies elípticas y el décimo problema de Hilbert

• Pastén, Héctor ^[1]
  1. [1] Universidad Católica de Chile, Departamento de Matemáticas, Pontificia Facultad de Matemáticas, Chile
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 30, Nº. 1, 2023 (Ejemplar dedicado a: Revista de Matemática: Teoría y Aplicaciones), págs. 113-120
• Idioma: español
• DOI: 10.15517/rmta.v30i1.52266
• Títulos paralelos:
  • Elliptic surfaces and Hilbert’s tenth problem
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Es sabido que se obtendría una solución negativa al décimo problema de Hilbert para el anillo de enteros OF de un campo de números F si Z fuera diofantino en OF. Denef y Lipshitz conjeturaron que esto último ocurre para todo F. En esta nota se demuestra que la conjetura de Denef y Lipshitz es consecuencia de una conocida conjetura sobre superficies elípticas.

  • English

    A negative solution to Hilbert’s tenth problem for the ring of integers OF of a number field F would follow if Z were Diophantine in OF. Denef and Lipshitz conjectured that the latter occurs for every number field F. In this note we show that the conjecture of Denef and Lipshitz is a consequence of a well-known conjecture on elliptic surfaces.

• Referencias bibliográficas
  • B. Conrad, K. Conrad, H. Helfgott, Root numbers and ranks in positive characteristic, Adv. Math. 198(2005), no. 2, 684-731. Doi: 10.48550/arXiv.math/0408153
  • G. Cornelissen, T. Pheidas, K. Zahidi, Division-ample sets and the Diophantine problem for rings of integers, J. Théor. Nombres Bordeaux 17(2005),...
  • M. Davis, H. Putnam, J. Robinson, The decision problem for exponential diophantine equations, Ann. of Math (2) 74(1961), no. 3, 425-436. Doi:...
  • J. Denef, Hilbert’s tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48(1975), no. 1, 214-220. Doi: 10.2307/2040720
  • J. Denef, Diophantine sets over algebraic integer rings II, Trans. Amer. Math. Soc. 257(1980), no. 1, 227-236. Doi: 10.2307/1998133
  • J. Denef, L. Lipshitz, Diophantine sets over some rings of algebraic integers, J. London Math. Soc. (2) 18(1978), no. 3, 385-391. Doi: 10.1112/jlms/s2-...
  • N. Garcia-Fritz, H. Pasten, Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points. Math. Ann. 377(2020),...
  • D. Kundu, A. Lei, F. Sprung, Studying Hilbert’s 10th problem via explicit elliptic curves, arXiv: 2207.07021(2022). Doi: 10.48550/arXiv.2207.07021
  • S. Lang, A. Néron, Rational points of abelian varieties over function fields, Amer. J. Math. 81(1959), no.1, 95-118. Doi: 10.2307/2372851
  • Y. Matiyasevich, The Diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR 191(1970), no. 2, 279-282.
  • B. Mazur, K. Rubin, Ranks of twists of elliptic curves and Hilbert’s tenth problem. Invent. Math. 181(2010), no. 3, 541-575. Doi: 10.1007/s00222-...
  • B. Mazur, K. Rubin, Diophantine stability, With an appendix by Michael Larsen. Amer. J. Math. 140(2018), no. 3, 571-616. Doi: 10.1353/ajm.2018.0014
  • B. Mazur, K. Rubin, A. Shlapentokh, Existential definability and Diophantine stability, arXiv: 2208.09963 (2022), Doi: 10.48550/arXiv.2208.09963
  • M. Murty, H. Pasten, Elliptic curves, L-functions, and Hilbert’s tenth problem. J. Number Theory 182(2018), 1-18. Doi: 10.1016/j.jnt.2017.07.008
  • A. Néron, Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps. Bull. Soc. Math. France...
  • T. Pheidas, Hilbert’s tenth problem for a class of rings of algebraic integers. Proc. Amer. Math. Soc. 104(1988), no. 2, 611-620. Doi: 10.2307/2047021
  • B. Poonen, Using elliptic curves of rank one towards the undecidability of Hilbert’s tenth problem over rings of algebraic integers. Fieker,...
  • A. Ray, Remarks on Hilbert’s tenth problem and the Iwasawa theory of elliptic curves, Bulletin of the Australian Mathematical Society Society...
  • C. Salgado, On the rank of the fibers of rational elliptic surfaces, Algebra Number Theory 6(2012), no. 7, 1289-1314. Doi: 10.2140/ant.2012.6.1289
  • C. Schwartz, An elliptic surface of Mordell-Weil rank 8 over the rational numbers, J. Théor. Nombres Bordeaux 6(1994) , no. 1, 1-8. Available...
  • A. Shlapentokh, Extension of Hilbert’s tenth problem to some algebraic number fields. Comm. Pure Appl. Math. 42(1989) , no. 7, 939-962. Doi:...
  • A. Shlapentokh, Elliptic curves retaining their rank in finite extensions and Hilbert’s tenth problem for rings of algebraic numbers, Trans....
  • J. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math 342(1983), 197-211. Doi: 10.1515/crll.1983.342.197
  • C. Videla, (1989). Sobre el décimo problema de Hilbert, Atas da Xa Escola de Algebra, Vitoria, ES, Brasil, Colecao Atas 16 Sociedade Brasileira...