Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of

Author:
Bjorn Poonen

Journal:
J. Amer. Math. Soc. **16** (2003), 981-990

MSC (2000):
Primary 11U05; Secondary 11G05

Published electronically:
July 8, 2003

MathSciNet review:
1992832

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[Aya92]**Mohamed Ayad,*Points 𝑆-entiers des courbes elliptiques*, Manuscripta Math.**76**(1992), no. 3-4, 305–324 (French). MR**1185022**, 10.1007/BF02567763**[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) Contemp. Math., vol. 270, Amer. Math. Soc., Providence, RI, 2000, pp. 253–260. MR**1802017**, 10.1090/conm/270/04377**[DLPVG00]**Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel (eds.),*Hilbert’s tenth problem: relations with arithmetic and algebraic geometry*, Contemporary Mathematics, vol. 270, American Mathematical Society, Providence, RI, 2000. Papers from the workshop held at Ghent University, Ghent, November 2–5, 1999. MR**1802007****[DPR61]**Martin Davis, Hilary Putnam, and Julia Robinson,*The decision problem for exponential diophantine equations*, Ann. of Math. (2)**74**(1961), 425–436. MR**0133227****[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]**Ju. V. Matijasevič,*The Diophantineness of enumerable sets*, Dokl. Akad. Nauk SSSR**191**(1970), 279–282 (Russian). MR**0258744****[Maz92]**Barry Mazur,*The topology of rational points*, Experiment. Math.**1**(1992), no. 1, 35–45. MR**1181085****[Maz95]**B. Mazur,*Speculations about the topology of rational points: an update*, Astérisque**228**(1995), 4, 165–182. Columbia University Number Theory Seminar (New York, 1992). MR**1330932****[Ser72]**Jean-Pierre Serre,*Propriétés galoisiennes des points d’ordre fini des courbes elliptiques*, Invent. Math.**15**(1972), no. 4, 259–331 (French). MR**0387283****[Ser73]**J.-P. Serre,*A course in arithmetic*, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR**0344216****[Ser81]**Jean-Pierre Serre,*Quelques applications du théorème de densité de Chebotarev*, Inst. Hautes Études Sci. Publ. Math.**54**(1981), 323–401 (French). MR**644559****[Ser97]**Jean-Pierre Serre,*Lectures on the Mordell-Weil theorem*, 3rd ed., Aspects of Mathematics, 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**1757192****[Shl94]**Alexandra Shlapentokh,*Diophantine classes of holomorphy rings of global fields*, J. Algebra**169**(1994), no. 1, 139–175. MR**1296586**, 10.1006/jabr.1994.1276**[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**1465332**, 10.1007/s002220050170**[Shl00]**Alexandra Shlapentokh,*Defining integrality at prime sets of high density in number fields*, Duke Math. J.**101**(2000), no. 1, 117–134. MR**1733734**, 10.1215/S0012-7094-00-10115-9**[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**1924099****[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**961918**, 10.1016/0022-314X(88)90019-4**[Sil92]**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. MR**1329092****[Vin54]**I. M. Vinogradov,*The method of trigonometrical sums in the theory of numbers*, Interscience Publishers, London and New York., no year given. Translated, revised and annotated by K. F. Roth and Anne Davenport. MR**0062183****[War48]**Morgan Ward,*Memoir on elliptic divisibility sequences*, Amer. J. Math.**70**(1948), 31–74. MR**0023275**

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