Counting special points: Logic, diophantine geometry, and transcendence theory

Scanlon, Thomas

doi:10.1090/S0273-0979-2011-01354-4

Counting special points: Logic, diophantine geometry, and transcendence theory
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Bull. Amer. Math. Soc. 49 (2012), 51-71 Request permission

Abstract:

We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier.

References

Yves André, $G$-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989. MR 990016, DOI 10.1007/978-3-663-14108-2
Yves André, Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math. 505 (1998), 203–208 (French, with English summary). MR 1662256, DOI 10.1515/crll.1998.118
James Ax, On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252–268. MR 277482, DOI 10.2307/1970774
E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), no. 2, 337–357. MR 1016893, DOI 10.1215/S0012-7094-89-05915-2
Enrico Bombieri, The Mordell conjecture revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 615–640. MR 1093712
Sinnou David, Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64 (1997), no. 1, 104–129 (French, with English and French summaries). MR 1450488, DOI 10.1006/jnth.1997.2100
Bas Edixhoven, Special points on the product of two modular curves, Compositio Math. 114 (1998), no. 3, 315–328. MR 1665772, DOI 10.1023/A:1000539721162
Bas Edixhoven and Andrei Yafaev, Subvarieties of Shimura varieties, Ann. of Math. (2) 157 (2003), no. 2, 621–645. MR 1973057, DOI 10.4007/annals.2003.157.621
A. M. Gabrièlov, Projections of semianalytic sets, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 18–30 (Russian). MR 0245831
M. Gromov, Entropy, homology and semialgebraic geometry, Astérisque 145-146 (1987), 5, 225–240. Séminaire Bourbaki, Vol. 1985/86. MR 880035
Alexandre Grothendieck, Esquisse d’un programme, Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 5–48 (French, with French summary). With an English translation on pp. 243–283. MR 1483107
Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR 1333294, DOI 10.1090/S0894-0347-96-00202-0
Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115. MR 1854232, DOI 10.1016/S0168-0072(01)00096-3
A. G. Khovanskiĭ, Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Smilka Zdravkovska. MR 1108621, DOI 10.1090/mmono/088
Julia F. Knight, Anand Pillay, and Charles Steinhorn, Definable sets in ordered structures. II, Trans. Amer. Math. Soc. 295 (1986), no. 2, 593–605. MR 833698, DOI 10.1090/S0002-9947-1986-0833698-1
Maxim Kontsevich and Don Zagier, Periods, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771–808. MR 1852188
Serge Lang, Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0214547
Henry B. Mann, On linear relations between roots of unity, Mathematika 12 (1965), 107–117. MR 191892, DOI 10.1112/S0025579300005210
D. Masser and U. Zannier. Torsion points on families of squares of elliptic curves. Mathematische Annalen, 16 February 2011. Online First.
D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132 (2010), no. 6, 1677–1691. MR 2766181
David Masser and Umberto Zannier, Torsion anomalous points and families of elliptic curves, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 491–494 (English, with English and French summaries). MR 2412783, DOI 10.1016/j.crma.2008.03.024
B. Mazur, Questions of decidability and undecidability in number theory, J. Symbolic Logic 59 (1994), no. 2, 353–371. MR 1276620, DOI 10.2307/2275395
Ya’acov Peterzil and Sergei Starchenko, Uniform definability of the Weierstrass $\wp$ functions and generalized tori of dimension one, Selecta Math. (N.S.) 10 (2004), no. 4, 525–550. MR 2134454, DOI 10.1007/s00029-005-0393-y
Ya’acov Peterzil and Sergei Starchenko. Around Pila-Zannier: the semiabelian case. preprint, 2009.
Ya’acov Peterzil and Sergei Starchenko. Definability of restricted theta functions and families of abelian varieties. preprint, 2010.
J. Pila and A. J. Wilkie, The rational points of a definable set, Duke Math. J. 133 (2006), no. 3, 591–616. MR 2228464, DOI 10.1215/S0012-7094-06-13336-7
Jonathan Pila, Rational points on a subanalytic surface, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1501–1516 (English, with English and French summaries). MR 2172272
Jonathan Pila, On the algebraic points of a definable set, Selecta Math. (N.S.) 15 (2009), no. 1, 151–170. MR 2511202, DOI 10.1007/s00029-009-0527-8
Jonathan Pila, Counting rational points on a certain exponential-algebraic surface, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 489–514 (English, with English and French summaries). MR 2667784
Jonathan Pila. O-minimality and the André-Oort conjecture for C${}^n$. Ann. of Math. (2), 172(3):1779–1840, 2011.
Jonathan Pila and Umberto Zannier, Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19 (2008), no. 2, 149–162. MR 2411018, DOI 10.4171/RLM/514
Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697, DOI 10.1090/S0002-9947-1986-0833697-X
Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. III, Trans. Amer. Math. Soc. 309 (1988), no. 2, 469–476. MR 943306, DOI 10.1090/S0002-9947-1988-0943306-9
M. Raynaud, Sous-variétés d’une variété abélienne et points de torsion, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 327–352 (French). MR 717600
J.-P. Rolin, P. Speissegger, and A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), no. 4, 751–777. MR 1992825, DOI 10.1090/S0894-0347-03-00427-2
Thomas Scanlon. A proof of the André-Oort conjecture via mathematical logic [after J. Pila, A. Wilkie and U. Zannier]. 2011. Séminaire Bourbaki. Vol. 2010/2011.
Thomas Scanlon. Theorems on unlikely intersections by counting points in definable sets. 2011. Mini-courses around the Pink-Zilber conjecture at Luminy, May 2011.
C. L. Siegel. Über die Classenzahl quadratischer Zahlkörper. Acta Arith., (1):83–86, 1935.
Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI 10.1515/crll.1983.342.197
Patrick Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189–211. MR 1676876, DOI 10.1515/crll.1999.026
Alfred Tarski, A Decision Method for Elementary Algebra and Geometry, The Rand Corporation, Santa Monica, Calif., 1948. MR 0028796
J. Tsimermann. Brauer-Siegel for arithmetic tori and lower bounds for galois orbits of special points. preprint, 2011.
E. Ullmo and A. Yafev. Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciaux. preprint, 2011.
Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
Lou van den Dries, Angus Macintyre, and David Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), no. 1, 183–205. MR 1289495, DOI 10.2307/2118545
Lou van den Dries and Chris Miller, On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994), no. 1-3, 19–56. MR 1264338, DOI 10.1007/BF02758635
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), no. 4, 1051–1094. MR 1398816, DOI 10.1090/S0894-0347-96-00216-0
Alex J. Wilkie, o-minimal structures, Astérisque 326 (2009), Exp. No. 985, vii, 131–142 (2010). Séminaire Bourbaki. Vol. 2007/2008. MR 2605320
Y. Yomdin, $C^k$-resolution of semialgebraic mappings. Addendum to: “Volume growth and entropy”, Israel J. Math. 57 (1987), no. 3, 301–317. MR 889980, DOI 10.1007/BF02766216
Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285–300. MR 889979, DOI 10.1007/BF02766215

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Counting special points: Logic, diophantine geometry, and transcendence theory
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