A proof of the André-Oort conjecture via mathematical logic [after Pila, Wilkie and Zannier]
Séminaire Bourbaki Volume 2010/2011 Exposés 1027-1042. Avec table par noms d'auteurs de 1948/49 à 2009/10., Astérisque, no. 348 (2012), Exposé no. 1037, 17 p.
@incollection{AST_2012__348__299_0,
     author = {Scanlon, Thomas},
     title = {A proof of the {Andr\'e-Oort} conjecture via mathematical logic [after {Pila,} {Wilkie} and {Zannier]}},
     booktitle = {S\'eminaire Bourbaki Volume 2010/2011 Expos\'es 1027-1042. Avec table par noms d'auteurs de 1948/49 \`a 2009/10.},
     series = {Ast\'erisque},
     note = {talk:1037},
     pages = {299--315},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {348},
     year = {2012},
     mrnumber = {3051200},
     zbl = {1271.14030},
     language = {en},
     url = {http://archive.numdam.org/item/AST_2012__348__299_0/}
}
TY  - CHAP
AU  - Scanlon, Thomas
TI  - A proof of the André-Oort conjecture via mathematical logic [after Pila, Wilkie and Zannier]
BT  - Séminaire Bourbaki Volume 2010/2011 Exposés 1027-1042. Avec table par noms d'auteurs de 1948/49 à 2009/10.
AU  - Collectif
T3  - Astérisque
N1  - talk:1037
PY  - 2012
SP  - 299
EP  - 315
IS  - 348
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_2012__348__299_0/
LA  - en
ID  - AST_2012__348__299_0
ER  - 
%0 Book Section
%A Scanlon, Thomas
%T A proof of the André-Oort conjecture via mathematical logic [after Pila, Wilkie and Zannier]
%B Séminaire Bourbaki Volume 2010/2011 Exposés 1027-1042. Avec table par noms d'auteurs de 1948/49 à 2009/10.
%A Collectif
%S Astérisque
%Z talk:1037
%D 2012
%P 299-315
%N 348
%I Société mathématique de France
%U http://archive.numdam.org/item/AST_2012__348__299_0/
%G en
%F AST_2012__348__299_0
Scanlon, Thomas. A proof of the André-Oort conjecture via mathematical logic [after Pila, Wilkie and Zannier], dans Séminaire Bourbaki Volume 2010/2011 Exposés 1027-1042. Avec table par noms d'auteurs de 1948/49 à 2009/10., Astérisque, no. 348 (2012), Exposé no. 1037, 17 p. http://archive.numdam.org/item/AST_2012__348__299_0/

[1] Y. André - G-functions and geometry, Aspects of Mathematics, E13, Priedr. Vieweg & Sohn, 1989. | MR | Zbl

[2] Y. André, "Finitude des couples d'invariants modulaires singuliers sur une courbe algébrique plane non modulaire", J. reine angew. Math. 505 (1998), p. 203-208. | DOI | MR | Zbl

[3] E. Bombieri - "The Mordell conjecture revisited", Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), p. 615-640. | EuDML | Numdam | MR | Zbl

[4] E. Bombieri & J. Pila - "The number of integral points on arcs and ovals", Duke Math. J. 59 (1989), p. 337-357. | DOI | MR | Zbl

[5] L. Butler - "Some cases of Wilkie's conjecture", preprint, 2011. | MR | Zbl

[6] C. Daw & A. Yafaev - "An unconditional proof of the André-Oort conjecture for Hilbert modular surfaces", Manuscripta Mathematica 135 (2011), p. 263-271. | DOI | MR | Zbl

[7] J. Denef & L. Van Den Dries - " p -adic and real subanalytic sets", Ann. of Math. 128 (1988), p. 79-138. | DOI | MR | Zbl

[8] L. Van Den Dries - "Remarks on Tarski's problem concerning ( 𝐑 , + , · , exp ) ", in Logic colloquium '82 (Florence, 1982), Stud. Logic Found. Math., vol. 112, North-Holland, 1984, p. 97-121 | DOI | MR | Zbl

[9] L. Van Den Dries, "A generalization of the Tarski-Seidenberg theorem, and some nondefinability results", Bull. Amer. Math. Soc. (N.S.) 15 (1986), p. 189-193. | DOI | MR | Zbl

[10] L. Van Den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge Univ. Press, 1998. | MR | Zbl

[11] L. Van Den Dries, A. Macintyre & D. Marker - "The elementary theory of restricted analytic fields with exponentiation", Ann. of Math. 140 (1994), p. 183-205. | DOI | MR | Zbl

[12] L. Van Den Dries & C. Miller - "On the real exponential field with restricted analytic functions", Israel J. Math. 85 (1994), p. 19-56. | DOI | MR | Zbl

[13] B. Edixhoven - "Special points on the product of two modular curves", Compositio Math. 114 (1998), p. 315-328. | DOI | MR | Zbl

[14] B. Edixhoven, "Special points on products of modular curves", Duke Math. J. 126 (2005), p. 325-348. | DOI | MR | Zbl

[15] B. Edixhoven & A. Yafaev - "Subvarieties of Shimura varieties", Ann. of Math. 157 (2003), p. 621-645. | DOI | MR | Zbl

[16] A. M. Gabrièlov - "Projections of semianalytic sets", Funkcional. Anal, i Priložen. 2 (1968), p. 18-30 (in Russian) | MR | Zbl

A. M. Gabrièlov - "Projections of semianalytic sets" ; English translation: Functional Anal. Appl. 2 (1968), 282-291. | DOI | Zbl

[17] M. Gromov - "Entropy, homology and semialgebraic geometry (after Y. Yomdin)", Séminaire Bourbaki, vol. 1985/86, exp. n° 663 Astérisque 145-146 (1987), p. 225-240. | EuDML | Numdam | MR | Zbl

[18] P. Habegger & J. Pila - "Some unlikely intersections beyond André-Oort", preprint, 2010. | MR | Zbl

[19] G. O. Jones, D. J. Miller & M. E. M. Thomas - "Mildness and the density of rational points on certain transcendental curves", Notre Dame J. Form. Log. 52 (2011), p. 67-74. | DOI | MR | Zbl

[20] G. O. Jones & M. E. M. Thomas - "The density of algebraic points on certain Pfaffian surfaces", Quarterly J. of Math. (2011), doi : 10.1093/qmath/harOll. | MR | Zbl

[21] A. G. Khovanskiĭ - Fewnomials, Translations of Mathematical Monographs, vol. 88, Amer. Math. Soc., 1991. | DOI | MR | Zbl

[22] B. Klingler & A. Yafaev - "The André-Oort conjecture", preprint, 2006. | MR | Zbl

[23] J. F. Knight, A. Pillay & C. Steinhorn - "Definable sets in ordered structures. II", Trans. Amer. Math. Soc. 295 (1986), p. 593-605. | DOI | MR | Zbl

[24] D. Masser & U. Zannier - "Torsion anomalous points and families of elliptic curves", C. R. Math. Acad. Sci. Paris 346 (2008), p. 491-494. | DOI | MR | Zbl

[25] D. Masser & U. Zannier, "Torsion anomalous points and families of elliptic curves", Amer. J. Math. 132 (2010), p. 1677-1691. | MR | Zbl

[26] D. Masser & U. Zannier, "Torsion points on families of squares of elliptic curves", Math. Ann. 352 (2012), p. 453-484. | DOI | MR | Zbl

[27] B. Moonen - "Linearity properties of Shimura varieties. II", Compositio Math. 114 (1998), p. 3-35. | DOI | MR | Zbl

[28] R. Noot - "Correspondances de Hecke, action de Galois et la conjecture d'André-Oort (d'après Edixhoven et Yafaev)", Séminaire Bourbaki, vol. 2004/2005, exp. n° 942, Astérisque 307 (2006), p. 165-197. | EuDML | Numdam | MR | Zbl

[29] F. Oort - "Some questions in algebraic geometry", http://www.math.uu.nl/people/oort/, 1995.

[30] Y. Peterzil & S. Starchenko - "Uniform definability of the Weierstrass functions and generalized tori of dimension one", Selecta Math. (N.S.) 10 (2004), p. 525-550. | DOI | MR | Zbl

[31] Y. Peterzil & S. Starchenko, "Around Pila-Zannier: the semiabelian case", preprint, 2009.

[32] Y. Peterzil & S. Starchenko, "Definability of restricted theta functions and families of Abelian varieties", preprint, 2010. | MR | Zbl

[33] J. Pila - "Integer points on the dilation of a subanalytic surface", Q. J. Math. 55 (2004), p. 207-223. | DOI | MR | Zbl

[34] J. Pila, "On the algebraic points of a definable set", Selecta Math. (N.S.) 15 (2009), p. 151-170. | DOI | MR | Zbl

[35] J. Pila, "Counting rational points on a certain exponential-algebraic surface", Ann. Inst. Fourier (Grenoble) 60 (2010), p. 489-514. | DOI | EuDML | Numdam | MR | Zbl

[36] J. Pila, "O-minimality and the André-Oort conjecture for n ", Ann. of Math. 173 (2011), p. 1779-1840. | DOI | MR | Zbl

[37] J. Pila & A. J. Wilkie - "The rational points of a definable set", Duke Math. J. 133 (2006), p. 591-616. | DOI | MR | Zbl

[38] J. Pila & U. Zannier - "Rational points in periodic analytic sets and the Manin-Mumford conjecture", Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), p. 149-162. | DOI | MR | Zbl

[39] A. Pillay & C. Steinhorn - "Definable sets in ordered structures. I", Trans. Amer. Math. Soc. 295 (1986), p. 565-592. | DOI | MR | Zbl

[40] A. Pillay & C. Steinhorn, "Definable sets in ordered structures. Ill", Trans. Amer. Math. Soc. 309 (1988), p. 469-476. | DOI | MR | Zbl

[41] M. Raynaud - "Sous-variétés d'une variété abélienne et points de torsion", in Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser, 1983, p. 327-352. | DOI | MR | Zbl

[42] T. Scanlon - "Counting special points : Logic, diophantine geometry, and transcendence theory", Bull. Amer. Math. Soc. (N.S.) 49 (2012), p. 51-71. | DOI | MR | Zbl

[43] J-P. Serre - Cours d'arithmétique, Presses Universitaires de France, 1970. | MR | Zbl

[44] C. Siegel - "Über die Classenzahl quadratischer Zahlkörper", Acta Arith. 1 (1935), p. 83-86. | DOI | EuDML | JFM

[45] J. H. Silverman - The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer, 1992. | MR | Zbl

[46] P. Speissegger - "The Pfaffian closure of an o-minimal structure", J. reine angew. Math. 508 (1999), p. 189-211. | DOI | MR | Zbl

[47] A. Tarski - A Decision Method for Elementary Algebra and Geometry, RAND Corporation, Santa Monica, Calif., 1948. | MR | Zbl

[48] J. Tsimerman - "Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points", J. of the A.M.S. (2012), article electronically published on April 12, 2012. | MR | Zbl

[49] E. Ullmo & A. Yafaev - "The André-Oort conjecture for products of modular curves", in Arithmetic geometry, Clay Math. Proc., vol. 8, Amer. Math. Soc., 2009, p. 431-439. | MR | Zbl

[50] E. Ullmo & A. Yafaev, "Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites galoisiennes de points spéciaux", preprint, 2011. | MR | Zbl

[51] A. J. Wilkie - "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function", J. Amer. Math. Soc. 9 (1996), p. 1051-1094. | DOI | MR | Zbl

[52] A. J. Wilkie, "A theorem of the complement and some new o-minimal structures", Selecta Math. (N.S.) 5 (1999), p. 397-421. | DOI | MR | Zbl

[53] A. J. Wilkie, "o-minimal structures", Séminaire Bourbaki, vol. 2007/2008, exp. n° 985, Astérisque 326 (2009), p. 131-142. | Numdam | MR | Zbl

[54] Y. Yomdin - " C k -resolution of semialgebraic mappings. Addendum to: "Volume growth and entropy"", Israel J. Math. 57 (1987), p. 301-317. | DOI | MR | Zbl

[55] Y. Yomdin, "Volume growth and entropy", Israel J. Math. 57 (1987), p. 285-300. | DOI | MR | Zbl