Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of
Author:
Bjorn Poonen
Journal:
J. Amer. Math. Soc. 16 (2003), 981990
MSC (2000):
Primary 11U05; Secondary 11G05
Published electronically:
July 8, 2003
MathSciNet review:
1992832
Fulltext PDF Free Access
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Abstract: We give the first examples of infinite sets of primes such that Hilbert's Tenth Problem over has a negative answer. In fact, we can take to be a density 1 set of primes. We show also that for some such there is a punctured elliptic curve over such that the topological closure of in has infinitely many connected components.
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 [Aya92]
 Mohamed Ayad, Points entiers des courbes elliptiques, Manuscripta Math. 76 (1992), no. 34, 305324.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. 253260. 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 25, 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), 425436.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. 242248.
 [Mat70]
 Yuri V. Matijasevic, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279282. MR 41:3390
 [Maz92]
 Barry Mazur, The topology of rational points, Experiment. Math. 1 (1992), no. 1, 3545. MR 93j:14020
 [Maz95]
 Barry Mazur, Speculations about the topology of rational points: an update, Astérisque (1995), no. 228, 4, 165182, Columbia University Number Theory Seminar (New York, 1992). MR 96c:11068
 [Ser72]
 JeanPierre Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259331. MR 52:8126
 [Ser73]
 JeanPierre Serre, A course in arithmetic, SpringerVerlag, New York, 1973, Translated from the French, Graduate Texts in Mathematics, No. 7. MR 49:8956
 [Ser81]
 JeanPierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. (1981), no. 54, 323401.MR 83k:12011
 [Ser97]
 JeanPierre Serre, Lectures on the MordellWeil 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, 139175. 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, 489507. MR 98h:11163
 [Shl00]
 Alexandra Shlapentokh, Defining integrality at prime sets of high density in number fields, Duke Math. J. 101 (2000), no. 1, 117134. 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, 227252. 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, 411422.
 [Sil88]
 Joseph H. Silverman, Wieferich's criterion and the conjecture, J. Number Theory 30 (1988), no. 2, 226237. MR 89m:11027
 [Sil92]
 Joseph H. Silverman, The arithmetic of elliptic curves, SpringerVerlag, 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), 3174. MR 9:332j
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Additional Information
Bjorn Poonen
Affiliation:
Department of Mathematics, University of California, Berkeley, California 947203840
Email:
poonen@math.berkeley.edu
DOI:
http://dx.doi.org/10.1090/S0894034703004338
PII:
S 08940347(03)004338
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 DMS0301280 and a Packard Fellowship.
Article copyright:
© Copyright 2003
American Mathematical Society
