Palabras clave
Anillos de enteros
Superficies elípticas
Curvas elípticas
Hilbert’s tenth problem
Rings of integers
Elliptic surfaces
Elliptic curves
Cómo citar
Resumen
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.
Citas
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), no. 3, 727-735. Doi: 10.5802/jtnb.516
M. Davis, H. Putnam, J. Robinson, The decision problem for exponential diophantine equations, Ann. of Math (2) 74(1961), no. 3, 425-436. Doi: 10.2307/1970289
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- 18.3.385
N. Garcia-Fritz, H. Pasten, Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points. Math. Ann. 377(2020), no. 3-4, 989-1013. Doi: 10.1007/s00208-020-01991-w
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- 010-0252-0
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 80(1952), 101-166. Doi: 10.24033/bsmf.1427
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, Claus and Kohel, David R (Eds.), Algorithmic Number Theory, Sidney, 2002, 33-42. Doi: 10.1007/3-540-45455-1_4
A. Ray, Remarks on Hilbert’s tenth problem and the Iwasawa theory of elliptic curves, Bulletin of the Australian Mathematical Society Society (2022), 1-11 Doi: 10.1017/S000497272200082X
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 from: http://www.numdam.org/item/JTNB_1994__6_1_1_0.pdf [21] H. Shapiro, A. Shlapentokh, Diophantine relationships between algebraic number fields, Comm. Pure Appl. Math. 42(1989), no. 8, 1113-1122. Doi: 10.1002/cpa.3160420805
A. Shlapentokh, Extension of Hilbert’s tenth problem to some algebraic number fields. Comm. Pure Appl. Math. 42(1989) , no. 7, 939-962. Doi: 10.1002/cpa.3160420703
A. Shlapentokh, Elliptic curves retaining their rank in finite extensions and Hilbert’s tenth problem for rings of algebraic numbers, Trans. Amer. Math. Soc. 360(2008), no. 7, 3541-3555. Doi: 10.1090/S0002-9947-08-04302-X
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 de Matematica, 95-108
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