Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of $\mathbb{Q}$


Author: Bjorn Poonen
Journal: J. Amer. Math. Soc. 16 (2003), 981-990
MSC (2000): Primary 11U05; Secondary 11G05
DOI: https://doi.org/10.1090/S0894-0347-03-00433-8
Published electronically: July 8, 2003
MathSciNet review: 1992832
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give the first examples of infinite sets of primes $S$such that Hilbert's Tenth Problem over $\mathbb{Z}[S^{-1}]$ has a negative answer. In fact, we can take $S$ to be a density 1 set of primes. We show also that for some such $S$ there is a punctured elliptic curve $E'$ over $\mathbb{Z}[S^{-1}]$such that the topological closure of $E'(\mathbb{Z}[S^{-1}])$ in $E'(\mathbb{R})$has infinitely many connected components.


References [Enhancements On Off] (What's this?)

  • [Aya92] Mohamed Ayad, Points $S$-entiers des courbes elliptiques, Manuscripta Math. 76 (1992), no. 3-4, 305-324.MR 93i:11064
  • [CZ00] Gunther Cornelissen and Karim Zahidi, Topology of Diophantine sets: remarks on Mazur's conjectures, Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math. Soc., Providence, RI, 2000, pp. 253-260. MR 2001m:11217
  • [DLPVG00] Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel (eds.), Hilbert's tenth problem: relations with arithmetic and algebraic geometry, American Mathematical Society, Providence, RI, 2000, Papers from the workshop held at Ghent University, Ghent, November 2-5, 1999. MR 2001g:00018
  • [DPR61] Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961), 425-436.MR 24:A3061
  • [Eve02] Graham Everest, Zsigmondy's theorem for elliptic curves, preprint, 11 October 2002.
  • [KR92] Ki Hang Kim and Fred W. Roush, An approach to rational Diophantine undecidability, Proceedings of Asian Mathematical Conference, 1990 (Hong Kong, 1990) (River Edge, NJ), World Sci. Publishing, 1992, pp. 242-248.
  • [Mat70] Yuri V. Matijasevic, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279-282. MR 41:3390
  • [Maz92] Barry Mazur, The topology of rational points, Experiment. Math. 1 (1992), no. 1, 35-45. MR 93j:14020
  • [Maz95] Barry Mazur, Speculations about the topology of rational points: an update, Astérisque (1995), no. 228, 4, 165-182, Columbia University Number Theory Seminar (New York, 1992). MR 96c:11068
  • [Ser72] Jean-Pierre Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259-331. MR 52:8126
  • [Ser73] Jean-Pierre Serre, A course in arithmetic, Springer-Verlag, New York, 1973, Translated from the French, Graduate Texts in Mathematics, No. 7. MR 49:8956
  • [Ser81] Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. (1981), no. 54, 323-401.MR 83k:12011
  • [Ser97] Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, third ed., Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre.MR 2000m:11049
  • [Shl94] Alexandra Shlapentokh, Diophantine classes of holomorphy rings of global fields, J. Algebra 169 (1994), no. 1, 139-175. MR 95h:12007
  • [Shl97] Alexandra Shlapentokh, Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator, Invent. Math. 129 (1997), no. 3, 489-507. MR 98h:11163
  • [Shl00] Alexandra Shlapentokh, Defining integrality at prime sets of high density in number fields, Duke Math. J. 101 (2000), no. 1, 117-134. MR 2001a:11200
  • [Shl02] Alexandra Shlapentokh, Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2, J. Number Theory 95 (2002), no. 2, 227-252. MR 2003h:03068
  • [Shl03] Alexandra Shlapentokh, A ring version of Mazur's conjecture on topology of rational points, Internat. Math. Res. Notices (2003), no. 7, 411-422.
  • [Sil88] Joseph H. Silverman, Wieferich's criterion and the $abc$-conjecture, J. Number Theory 30 (1988), no. 2, 226-237. MR 89m:11027
  • [Sil92] Joseph H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original. MR 95m:11054
  • [Vin54] I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Interscience Publishers, London and New York, 1954, Translated, revised and annotated by K. F. Roth and Anne Davenport. MR 15:941b
  • [War48] Morgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31-74. MR 9:332j

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11U05, 11G05

Retrieve articles in all journals with MSC (2000): 11U05, 11G05


Additional Information

Bjorn Poonen
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: poonen@math.berkeley.edu

DOI: https://doi.org/10.1090/S0894-0347-03-00433-8
Keywords: Hilbert's Tenth Problem, elliptic curve, Mazur's Conjecture, diophantine definition
Received by editor(s): December 8, 2002
Published electronically: July 8, 2003
Additional Notes: This research was supported by NSF grant DMS-0301280 and a Packard Fellowship.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society