Model theory and diophantine geometry

Pillay, Anand

doi:10.1090/S0273-0979-97-00730-1

Model theory and diophantine geometry
HTML articles powered by AMS MathViewer

by Anand Pillay PDF

Bull. Amer. Math. Soc. 34 (1997), 405-422 Request permission

Erratum: Bull. Amer. Math. Soc. 35 (1998), 67.

Abstract:

I discuss some recent applications of model theory to diophantine-type problems in algebraic geometry. I give the required background, as well as a sketch of the proofs.

References

Dan Abramovich and José Felipe Voloch, Toward a proof of the Mordell-Lang conjecture in characteristic $p$, Internat. Math. Res. Notices 5 (1992), 103–115. MR 1162230, DOI 10.1155/S1073792892000126
A. Buium, Intersections in jet spaces and a conjecture of S. Lang, Ann. of Math. (2) 136 (1992), no. 3, 557–567. MR 1189865, DOI 10.2307/2946600
Alexandru Buium, Effective bound for the geometric Lang conjecture, Duke Math. J. 71 (1993), no. 2, 475–499. MR 1233446, DOI 10.1215/S0012-7094-93-07120-7
Alexandru Buium, Uniform bound for generic points of curves in tori, J. Reine Angew. Math. 469 (1995), 211–219. MR 1363831, DOI 10.1515/crll.1995.469.211
Alexandru Buium, Differential algebraic groups of finite dimension, Lecture Notes in Mathematics, vol. 1506, Springer-Verlag, Berlin, 1992. MR 1176753, DOI 10.1007/BFb0087235
A. Buium and A. Pillay, A gap theorem for abelian varieties over differential fields, Math. Research Letters 4 (1997), 211-219.
Z. Chatzidakis and E. Hrushovski, Model theory of difference fields, preprint 1995.
Robert F. Coleman, Manin’s proof of the Mordell conjecture over function fields, Enseign. Math. (2) 36 (1990), no. 3-4, 393–427. MR 1096426
Phyllis Joan Cassidy, The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras, J. Algebra 121 (1989), no. 1, 169–238. MR 992323, DOI 10.1016/0021-8693(89)90092-6
Ching-Li Chai, A note on Manin’s theorem of the kernel, Amer. J. Math. 113 (1991), no. 3, 387–389. MR 1109343, DOI 10.2307/2374831
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
Gerd Faltings, The general case of S. Lang’s conjecture, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 175–182. MR 1307396
Hans Grauert, Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 131–149 (German). MR 222087, DOI 10.1007/BF02684399
Marc Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575–603 (French). MR 969244, DOI 10.1007/BF01394276
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
E. Hrushovski, Difference fields and the Manin-Mumford conjecture, preprint 1996.
E. Hrushovski and A. Pillay, Weakly normal groups, in Logic Colloquium ’85, North-Holland, 1987.
E. Hrushovski and Z. Sokolovic, Minimal sets in differentially closed fields, to appear in Trans. AMS.
Ehud Hrushovski and Boris Zilber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), no. 1, 1–56. MR 1311822, DOI 10.1090/S0894-0347-96-00180-4
Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
Serge Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229–234. MR 190146, DOI 10.1007/BF02410091
Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. MR 1112552, DOI 10.1007/978-3-642-58227-1
Angus Macintyre, Kenneth McKenna, and Lou van den Dries, Elimination of quantifiers in algebraic structures, Adv. in Math. 47 (1983), no. 1, 74–87. MR 689765, DOI 10.1016/0001-8708(83)90055-5
Yu. Manin, Rational points of algebraic curves over function fields, AMS Translations Ser. II 59 (1966), 189-234.
D. Marker, Model theory of differential fields, in Model Theory of Fields, D. Marker, M. Messmer and A. Pillay, Lecture Notes in Logic 5, Springer, 1996.
M. Messmer, Some model theory of separably closed fields, in Model Theory of Fields, D. Marker, M. Messmer and A. Pillay, Lecture Notes in Logic 5, Springer 1996.
Margit Messmer, Groups and fields interpretable in separably closed fields, Trans. Amer. Math. Soc. 344 (1994), no. 1, 361–377. MR 1231337, DOI 10.1090/S0002-9947-1994-1231337-6
A. Pillay, Geometric Stability Theory, Oxford University Press, 1996.
Michael McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995), no. 1, 143–159. MR 1323985, DOI 10.1007/BF01241125
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
Pierre Samuel, Compléments à un article de Hans Grauert sur la conjecture de Mordell, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 55–62 (French). MR 204430, DOI 10.1007/BF02684805
Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. MR 1002324, DOI 10.1007/978-3-663-14060-3
S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551