Model Theory of Adeles and Number Theory

References (89)

1. Ax, J. The elementary theory of finite fields. Ann. of Math. (2) 88 (1968), 239-271.
2. Ax, J., and Kochen, S. Diophantine problems over local fields. I. Amer. J. Math. 87 (1965), 605-630.
3. Ax, J., and Kochen, S. Diophantine problems over local fields. II. A complete set of axioms for p-adic number theory. Amer. J. Math. 87 (1965), 631-648.
4. Ax, J., and Kochen, S. Diophantine problems over local fields. III. Decidable fields. Ann. of Math. (2) 83 (1966), 437-456.
5. Bakker, Benjamin, K. B., and Tsimerman, J. Tame topology of arithmetic quotients and algebraicity of hodge loci. arXiv:1810.04801 (2018).
6. Basarab, Ş. A. Relative elimination of quantifiers for Henselian valued fields. Ann. Pure Appl. Logic 53, 1 (1991), 51-74.
7. Bélair, L. Substructures and uniform elimination for p-adic fields. Ann. Pure Appl. Logic 39, 1 (1988), 1-17.
8. Berman, M., Derakhshan, J., Onn, U., and Paajanen, P. Uniform cell decomposition and applications to Chevalley groups. Journal of London Mathematical Society, 87, 2 (2013), 586-606.
9. Bump, D. Automorphic forms and representations, vol. 55 of Cambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 1997.
10. Bump, D., Cogdell, J. W., de Shalit, E., Gaitsgory, D., Kowalski, E., and Kudla, S. S. An introduction to the Langlands program. Birkhäuser Boston, Inc., Boston, MA, 2003. Lectures presented at the Hebrew University of Jerusalem, Jerusalem, March 12-16, 2001, Edited by Joseph Bernstein and Stephen Gelbart.
11. Cassels, J. W. S. Global fields. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). Thompson, Washington, D.C., 1967, pp. 42-84.
12. Cassels, J. W. S. Local fields, vol. 3 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1986.
13. Cassels, J. W. S., and Fröhlich, A., Eds. Algebraic number theory (London, 1986), Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. Reprint of the 1967 original.
14. Chatzidakis, Z., van den Dries, L., and Macintyre, A. Definable sets over finite fields. J. Reine Angew. Math. 427 (1992), 107-135.
15. Cherlin, G. Model-Theoretic Algebra, vol. 521 of Lecture Notes in Mathematics. Springer- Verlag, 1976.
16. Cherlin, G. Model theoretic algebra-selected topics. Lecture Notes in Mathematics, Vol. 521. Springer-Verlag, Berlin-New York, 1976.
17. Chernikov, A., and Hils, M. Valued difference fields and N T P 2 . arXiv:12081341.
18. Cluckers, R., Derakhshan, J., Leenknegt, E., and Macintyre, A. Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields. Ann. Pure Appl. Logic 164, 12 (2013), 1236-1246.
19. Cluckers, R., and Loeser, F. Constructible motivic functions and motivic integration. Invent. Math. 173 (2008), 23-121.
20. Cluckers, R., and Loeser, F. Constructible exponential functions, motivic fourier trans- form, and transfer princple. Ann. Math 171, 2 (2010), 1011-1065.
21. Connes, A. Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Math. (N.S.) 5, 1 (1999), 29-106.
22. Connes, A., and Consani, C. The hyperring of adèle classes. J. Number Theory 131, 2 (2011), 159-194.
23. Connes, A., and Consani, C. The arithmetic site. C. R. Math. Acad. Sci. Paris 352, 12 (2014), 971-975.
24. Connes, A., and Consani, C. Geometry of the arithmetic site. Adv. Math. 291 (2016), 274-329.
25. D'Aquino, P., and Macintyre, A. Commutative unital rings elementarily equivalent to prescribed product products. Preprint (2019).
26. D'Aquino, P., and Macintyre, A. Zilber problem on residue rings of models of arithemtic, part 1: the prime power case. Preprint (2019).
27. D'Aquino Paola, Derakhshan, J., and Macintyre, A. Truncations of ordered abelian groups. Preprint (arXiv: 1910/14517).
28. Denef, J. The rationality of the Poincaré series associated to the p-adic points on a variety. Invent. Math. 77 (1984), 1 -23.
29. Denef, J., and Loeser, F. Definable sets, motives, and p-adic integrals. J. Amer. Math. Soc. 14, 2 (2001), 429-469.
30. Derakhshan, J. Euler products of p-adic integrals and definable sets I: meromorphic continuation and zeta functions of groups. In Preparation.
31. Derakhshan, J. Euler products of p-adic integrals and definable sets II: height zeta func- tions and equidistribution of rational points. In Preparation.
32. Derakhshan, J. p-adic model theory, p-adic integration, Euler products, and zeta functions of groups. this volume.
33. Derakhshan, J., and Macintyre, A. Some supplements to Feferman-Vaught related to the model theory of adeles. Ann. Pure Appl. Logic 165, 11 (2014), 1639-1679.
34. Derakhshan, J., and Macintyre, A. Enrichments of Boolean algebras by Presburger predicates. Fund. Math. 239, 1 (2017), 1-17.
35. Derakhshan, J., and Macintyre, A. Axioms for Commutative Unital Rings Elemen- tarily Equivalent to Restricted Products of Connected Rings. Preprint (2020).
36. Derakhshan, J., and Macintyre, A. Model completeness for Henselian valued fields with finite ramification valued in a Z-group. Preprint (arXiv: 1603.08598).
37. Derakhshan, J., and Macintyre, A. Model theory of adeles I.
38. Derakhshan, J., and Macintyre, A. Decidability problems for adele rings and related restricted products. Preprint (arXiv: 1910.14471).
39. Dougherty, R., and Miller, C. Definable Boolean combinations of open sets are Boolean combinations of open definable sets. Illinois J. Math. 45, 4 (2001), 1347-1350.
40. Duret, J.-L. Les corps faiblement algébriquement clos non séparablement clos ont la propriété d'indépendence. In Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), vol. 834 of Lecture Notes in Math. Springer, Berlin, 1980, pp. 136-162.
41. Enderton, H. B. A mathematical introduction to logic. Academic Press, New York- London, 1972.
42. Enderton, H. B. A mathematical introduction to logic, second ed. Harcourt/Academic Press, Burlington, MA, 2001.
43. Feferman, S., and Vaught, R. L. The first order properties of products of algebraic systems. Fund. Math. 47 (1959), 57-103.
44. Fried, M., and Jarden, M. Field arithmetic, vol. 3 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, 1986.
45. Fried, M., and Sacerdote, G. Solving Diophantine problems over all residue class fields of a number field and all finite fields. Ann. of Math. (2) 104, 2 (1976), 203-233.
46. Gelbart, S., and Shahidi, F. Analytic properties of automorphic L-functions, vol. 6 of Perspectives in Mathematics. Academic Press, Inc., Boston, MA, 1988.
47. Godement, R. Notes on Jacquet-Langlands' theory, vol. 8 of CTM. Classical Topics in Mathematics. Higher Education Press, Beijing, 2018. With commentaries by Robert Lang- lands and Herve Jacquet.
48. Godement, R., and Jacquet, H. Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260. Springer-Verlag, Berlin-New York, 1972.
49. Gorodnik, A., and Oh, H. Rational points on homogeneous varieties and equidistribution of adelic periods. Geom. Funct. Anal. 21, 2 (2011), 319-392. With an appendix by Mikhail Borovoi.
50. Hardy, G. H., and Wright, E. M. An introduction to the theory of numbers, sixth ed. Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles.
51. Haskell, D., Hrushovski, E., and Macpherson, D. Stable domination and indepen- dence in algebraically closed valued fields, vol. 30 of Lecture Notes in Logic. Association for Symbolic Logic, Chicago, IL, 2008.
52. Hrushovski, E. Ax's theorem with an additive character. Preprint (arXiv:1911.01096).
53. Hrushovski, E., and Kazhdan, D. Integration in valued fields. In Algebraic geometry and number theory, vol. 253 of Progr. Math. Birkhäuser Boston, Boston, MA, 2006, pp. 261-405.
54. Hrushovski, E., Martin, B., and Rideau, S. Definable equivalence relations and zeta functions of groups. J. Eur. Math. Soc. (JEMS) 20, 10 (2018), 2467-2537. With an appendix by Raf Cluckers.
55. Hrushovski, E., and Pillay, A. Groups definable in local fields and pseudo-finite fields. Israel J. Math. 85, 1-3 (1994), 203-262.
56. Iwasawa, K. On the rings of valuation vectors. Ann. of Math. (2) 57 (1953), 331-356.
57. Jacquet, H., and Langlands, R. P. Automorphic forms on GL(2). Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970.
58. Kiefe, C. Sets definable over finite fields: their zeta-functions. Trans. Amer. Math. Soc. 223 (1976), 45-59.
59. Kochen, S. The model theory of local fields. In 1xsy) ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974). Springer, Berlin, 1975, pp. 384-425. Lecture Notes in Math., Vol. 499.
60. Koenigsmann, J. Defining Z in Q. Ann. of Math. (2) 183, 1 (2016), 73-93.
61. Kontsevich, M., and Zagier, D. Periods. In Mathematics unlimited-2001 and beyond. Springer, Berlin, 2001, pp. 771-808.
62. Kreisel, G., and Krivine, J.-L. Elements of mathematical logic. Model theory. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1967.
63. Kuhlmann, F.-V. Quantifier elimination for Henselian fields relative to additive and mul- tiplicative congruences. Israel J. Math. 85, 1-3 (1994), 277-306.
64. Lafforgue, L. Le principe de fonctorialite de langlands comme un probleme de generali- sation de la loi d addition.
65. Lafforgue, L. Formules de Poisson non linéaires et principe de fonctorialité de Langlands. In Introduction to modern mathematics, vol. 33 of Adv. Lect. Math. (ALM). Int. Press, Somerville, MA, 2015, pp. 323-347.
66. Langlands, R. P. Problems in the theory of automorphic forms. In Lectures in modern analysis and applications, III. 1970, pp. 18-61. Lecture Notes in Math., Vol. 170.
67. Langlands, R. P. L-functions and automorphic representations. In Proceedings of the International Congress of Mathematicians (Helsinki, 1978) (1980), Acad. Sci. Fennica, Helsinki, pp. 165-175.
68. Macintyre, A. On definable subsets of p-adic fields. J. Symbolic Logic 41, 3 (1976), 605- 610.
69. Macintyre, A. Rationality of p-adic Poincaré series: uniformity in p. Ann. Pure Appl. Logic 49, 1 (1990), 31-74.
70. Manin, Y. I., and Panchishkin, A. A. Introduction to modern number theory, sec- ond ed., vol. 49 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. Fundamental problems, ideas and theories, Translated from the Russian.
71. Neukirch, J. Algebraic number theory, vol. 322 of Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences].
72. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.
73. Ono, T. An integral attached to a hypersurface. Amer. J. Math. 90 (1968), 1224-1236.
74. Pas, J. Uniform p-adic cell decomposition and local zeta functions. J. Reine Angew. Math. 399 (1989), 137-172.
75. Pas, J. Cell decomposition and local zeta functions in a tower of unramified extensions of a p-adic field. Proc. London Math. Soc. (3) 60, 1 (1990), 37-67.
76. Perlis, R. On the equation ζ K (s) = ζ K ′ (s). J. Number Theory 9, 3 (1977), 342-360.
77. Platonov, V., and Rapinchuk, A. Algebraic groups and number theory, vol. 139 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen.
78. Prestel, A., and Roquette, P. Formally p-adic fields, vol. 1050 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1984.
79. Ramakrishnan, D., and Valenza, R. J. Fourier analysis on number fields, vol. 186 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1999.
80. Roquette, P. The Riemann hypothesis in characteristic p in historical perspective, vol. 2222 of Lecture Notes in Mathematics. Springer, Cham, 2018. History of Mathematics Subseries.
81. Sarnak, P., Shin, S.-W., and Templier, N. Families of l-functions and their symmetry. Preprint (aXiv:1401.5507).
82. Serre, J.-P. Lectures on N X (p), vol. 11 of Chapman & Hall/CRC Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2012.
83. Tate, J. T. Fourier analysis in number fields, and Hecke's zeta-functions. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). Thompson, Washington, D.C., 1967, pp. 305-347.
84. Uchida, K. Isomorphisms of Galois groups. J. Math. Soc. Japan 28, 4 (1976), 617-620.
85. van den Dries, L., Macintyre, A., and Marker, D. The elementary theory of re- stricted analytic fields with exponentiation. Ann. of Math. (2) 140, 1 (1994), 183-205.
86. Weil, A. Adeles and algebraic groups, vol. 23 of Progress in Mathematics. Birkhäuser, Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono.
87. Weispfenning, V. Model theory of lattice products. Habilitation, Universitat Heidelberg (1978).
88. Weispfenning, V. Quantifier elimination and decision procedures for valued fields. In Models and sets (Aachen, 1983), vol. 1103 of Lecture Notes in Math. Springer, Berlin, 1984, pp. 419-472.
89. St Hilda's College, University of Oxford, Cowley Place, Oxford OX4 1DY, UK E-mail address: derakhsh@maths.ox.ac.uk