Diophantine definability and decidability in extensions of degree 2 of totally real fields

Colliot-Thélène, 1997, Double fibres and double covers: Paucity of rational points, Acta Arith., 79, 113, 10.4064/aa-79-2-113-135 Cornelissen, 2005, Division-ample sets and diophantine problem for rings of integers, J. Théor. Nombres Bordeaux, 17, 727, 10.5802/jtnb.516 Cornelissen, 2000, Topology of diophantine sets: Remarks on Mazur's conjectures, vol. 270, 253 Davis, 1973, Hilbert's tenth problem is unsolvable, Amer. Math. Monthly, 80, 233, 10.2307/2318447 Davis, 1976, Hilbert's tenth problem. Diophantine equations: Positive aspects of a negative solution, vol. 28, 323 Denef, 1975, Hilbert's tenth problem for quadratic rings, Proc. Amer. Math. Soc., 48, 214 Denef, 1980, Diophantine sets of algebraic integers, II, Trans. Amer. Math. Soc., 257, 227, 10.1090/S0002-9947-1980-0549163-X Denef, 1978, Diophantine sets over some rings of algebraic integers, J. London Math. Soc., 18, 385, 10.1112/jlms/s2-18.3.385 Lang, 1970 Mazur, 1992, The topology of rational points, Experiment. Math., 1, 35, 10.1080/10586458.1992.10504244 Mazur, 1994, Questions of decidability and undecidability in number theory, J. Symbolic Logic, 59, 353, 10.2307/2275395 Mazur, 1995, Speculation about the topology of rational points: An up-date, Asterisque, 228, 165 Mazur, 1998, Open problems regarding rational points on curves and varieties Pheidas, 1988, Hilbert's tenth problem for a class of rings of algebraic integers, Proc. Amer. Math. Soc., 104, 611 Bjorn Poonen, Elliptic curves whose rank does not grow and Hilbert's Tenth Problem over the rings of integers, private communication Poonen, 2002, Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers, vol. 2369, 33 Poonen, 2003, Hilbert's Tenth Problem and Mazur's conjecture for large subrings of Q, J. Amer. Math. Soc., 16, 981, 10.1090/S0894-0347-03-00433-8 Poonen, 2005, Diophantine definability of infinite discrete non-archimedean sets and diophantine models for large subrings of number fields, J. Reine Angew. Math., 2005, 27, 10.1515/crll.2005.2005.588.27 Serre, 1981, Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci., 54, 323, 10.1007/BF02698692 Shapiro, 1989, Diophantine relations between algebraic number fields, Comm. Pure Appl. Math., XLII, 1113, 10.1002/cpa.3160420805 Shlapentokh, 1989, Extension of Hilbert's tenth problem to some algebraic number fields, Comm. Pure Appl. Math., XLII, 939, 10.1002/cpa.3160420703 Shlapentokh, 1994, Diophantine classes of holomorphy rings of global fields, J. Algebra, 169, 139, 10.1006/jabr.1994.1276 Shlapentokh, 1994, Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of Q, Ann. Pure Appl. Logic, 68, 299, 10.1016/0168-0072(94)90024-8 Shlapentokh, 1997, Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator, Invent. Math., 129, 489, 10.1007/s002220050170 Shlapentokh, 2000, Hilbert's tenth problem over number fields, a survey, vol. 270, 107 Shlapentokh, 2002, On diophantine decidability and definability in some rings of algebraic functions of characteristic 0, J. Symbolic Logic, 67, 759, 10.2178/jsl/1190150109 Shlapentokh, 2002, On diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2, J. Number Theory, 95, 227, 10.1016/S0022-314X(01)92759-3 Shlapentokh, 2003, A ring version of Mazur's conjecture on topology of rational points, Int. Math. Res. Not., 2003, 411, 10.1155/S107379280320516X Shlapentokh, 2004, On diophantine definability and decidability in some infinite totally real extensions of Q, Trans. Amer. Math. Soc., 356, 3189, 10.1090/S0002-9947-03-03343-9 Shlapentokh, 2006 Alexandra Shlapentokh, Elliptic curves retaining their rank in finite extensions and Hilbert's tenth problem, Trans. Amer. Math. Soc., in press