Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of ℚ

Poonen, Bjorn

doi:10.1090/S0894-0347-03-00433-8

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
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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.

Mohamed Ayad, Points $S$-entiers des courbes elliptiques, Manuscripta Math. 76 (1992), no. 3-4, 305–324 (French). MR 1185022, DOI https://doi.org/10.1007/BF02567763
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, DOI https://doi.org/10.1090/conm/270/04377
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
Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961), 425–436. MR 133227, DOI https://doi.org/10.2307/1970289

[Eve02]everest2002 Graham Everest, Zsigmondy’s theorem for elliptic curves, preprint, 11 October 2002.

[KR92]kim-roush1992b 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.

Ju. V. Matijasevič, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279–282 (Russian). MR 0258744
Barry Mazur, The topology of rational points, Experiment. Math. 1 (1992), no. 1, 35–45. MR 1181085
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
Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI https://doi.org/10.1007/BF01405086
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
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
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
Alexandra Shlapentokh, Diophantine classes of holomorphy rings of global fields, J. Algebra 169 (1994), no. 1, 139–175. MR 1296586, DOI https://doi.org/10.1006/jabr.1994.1276
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, DOI https://doi.org/10.1007/s002220050170
Alexandra Shlapentokh, Defining integrality at prime sets of high density in number fields, Duke Math. J. 101 (2000), no. 1, 117–134. MR 1733734, DOI https://doi.org/10.1215/S0012-7094-00-10115-9
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]shlapentokh2003 Alexandra Shlapentokh, A ring version of Mazur’s conjecture on topology of rational points, Internat. Math. Res. Notices (2003), no. 7, 411–422.

Joseph H. Silverman, Wieferich’s criterion and the $abc$-conjecture, J. Number Theory 30 (1988), no. 2, 226–237. MR 961918, DOI https://doi.org/10.1016/0022-314X%2888%2990019-4
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]vinogradov1954 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.

[War48]ward1948 Morgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31–74.