Geodesic
Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of ℚ
Poonen, Bjorn1
1 Department of Mathematics, University of California, Berkeley, California 94720-3840
Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 981-990

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-03-00433-8

Poonen, Bjorn 1

1 Department of Mathematics, University of California, Berkeley, California 94720-3840 
@article{10_1090_S0894_0347_03_00433_8,
     author = {Poonen, Bjorn},
     title = {Hilbert\^a€™s {Tenth} {Problem} and {Mazur\^a€™s} {Conjecture} for large subrings of \^a„š},
     journal = {Journal of the American Mathematical Society},
     pages = {981--990},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2003},
     doi = {10.1090/S0894-0347-03-00433-8},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00433-8/}
}
TY  - JOUR
AU  - Poonen, Bjorn
TI  - Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of ℚ
JO  - Journal of the American Mathematical Society
PY  - 2003
SP  - 981
EP  - 990
VL  - 16
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00433-8/
DO  - 10.1090/S0894-0347-03-00433-8
ID  - 10_1090_S0894_0347_03_00433_8
ER  - 
%0 Journal Article
%A Poonen, Bjorn
%T Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of ℚ
%J Journal of the American Mathematical Society
%D 2003
%P 981-990
%V 16
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00433-8/
%R 10.1090/S0894-0347-03-00433-8
%F 10_1090_S0894_0347_03_00433_8
Poonen, Bjorn. Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of ℚ. Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 981-990. doi: 10.1090/S0894-0347-03-00433-8

[1] Ayad, Mohamed Points 𝑆-entiers des courbes elliptiques Manuscripta Math. 1992 305 324

[2] Cornelissen, Gunther, Zahidi, Karim Topology of Diophantine sets: remarks on Mazur’s conjectures 2000 253 260

[3] Hilbert’s tenth problem: relations with arithmetic and algebraic geometry 2000

[4] Davis, Martin, Putnam, Hilary, Robinson, Julia The decision problem for exponential diophantine equations Ann. of Math. (2) 1961 425 436

[5] Matijaseviä, Ju. V. The Diophantineness of enumerable sets Dokl. Akad. Nauk SSSR 1970 279 282

[6] Mazur, Barry The topology of rational points Experiment. Math. 1992 35 45

[7] Mazur, B. Speculations about the topology of rational points: an update Astérisque 1995

[8] Serre, Jean-Pierre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques Invent. Math. 1972 259 331

[9] Serre, J.-P. A course in arithmetic 1973

[10] Serre, Jean-Pierre Quelques applications du théorème de densité de Chebotarev Inst. Hautes Études Sci. Publ. Math. 1981 323 401

[11] Serre, Jean-Pierre Lectures on the Mordell-Weil theorem 1997

[12] Shlapentokh, Alexandra Diophantine classes of holomorphy rings of global fields J. Algebra 1994 139 175

[13] Shlapentokh, Alexandra Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator Invent. Math. 1997 489 507

[14] Shlapentokh, Alexandra Defining integrality at prime sets of high density in number fields Duke Math. J. 2000 117 134

[15] Shlapentokh, Alexandra Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2 J. Number Theory 2002 227 252

[16] Silverman, Joseph H. Wieferich’s criterion and the 𝑎𝑏𝑐-conjecture J. Number Theory 1988 226 237

[17] Silverman, Joseph H. The arithmetic of elliptic curves 1992

[18] Perlis, Sam Maximal orders in rational cyclic algebras of composite degree Trans. Amer. Math. Soc. 1939 82 96

[19] Birkhoff, Garrett, Ward, Morgan A characterization of Boolean algebras Ann. of Math. (2) 1939 609 610

Cité par Sources :