Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Model theory and diophantine geometry


Author: Anand Pillay
Journal: Bull. Amer. Math. Soc. 34 (1997), 405-422
MSC (1991): Primary 03C60, 14G05
DOI: https://doi.org/10.1090/S0273-0979-97-00730-1
Erratum: Bull. Amer. Math. Soc. 35 (1998), no. 1, 67 - 67.
MathSciNet review: 1458425
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. D. Abramovich and J. Voloch, Toward a proof of the Mordell-Lang conjecture in characteristic $p$, International Math. Res. Notices 5 (1992), 103-115. MR 94f:11051
  • 2. A. Buium, Intersections in jet spaces and a conjecture of S. Lang, Annals of Math. 136 (1992), 557-567. MR 93j:14055
  • 3. A. Buium, Effective bound for the geometric Lang conjecture, Duke Math. Journal 71 (1993), 475-499. MR 95c:14055
  • 4. A. Buium, Uniform bound for generic points of curves in tori, J. reine angew. Math. 469 (1995), 211-219. MR 96k:12012
  • 5. A. Buium, Differential algebraic groups of finite dimension, Springer Lecture Notes 1506, 1992. MR 93i:12010
  • 6. A. Buium and A. Pillay, A gap theorem for abelian varieties over differential fields, Math. Research Letters 4 (1997), 211-219.
  • 7. Z. Chatzidakis and E. Hrushovski, Model theory of difference fields, preprint 1995.
  • 8. R. Coleman, Manin's proof of the Mordell conjecture over function fields, L'enseignment Mathematique 36 (1990), 393-427. MR 92e:11069
  • 9. Ph. J. Cassidy, The classification of the semisimple differential algebraic groups, Journal of Algebra 121 (1989), 169-238. MR 90g:12007
  • 10. Ching-Li Chai, A note on Manin's theorem of the kernel, Amer. J. Math. 113 (1991), 387-389. MR 93b:14036
  • 11. G. Faltings, Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpern, Inventiones Math. 73 (1983), 349-366. MR 85g:11026a
  • 12. G. Faltings, The general case of S. Lang's conjecture, in Barsotti Symposium in Algebraic Geometry, Academic Press, 1994. MR 95m:11061
  • 13. H. Grauert, Mordell's Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper, Publ. Math. IHES 25 (1965), 131-149. MR 36:5139
  • 14. M. Hindry, Autour d'une conjecture de Serge Lang, Inventiones Math. 94 (1988), 575-603. MR 89k:11046
  • 15. E. Hrushovski, The Mordell-Lang conjecture for function fields, Journal of AMS 9 (1996), 667-690. MR 97h:11154
  • 16. E. Hrushovski, Difference fields and the Manin-Mumford conjecture, preprint 1996.
  • 17. E. Hrushovski and A. Pillay, Weakly normal groups, in Logic Colloquium '85, North-Holland, 1987.
  • 18. E. Hrushovski and Z. Sokolovic, Minimal sets in differentially closed fields, to appear in Trans. AMS.
  • 19. E. Hrushovski and B. Zilber, Zariski geometries, Journal of AMS 9 (1996), 1-56. MR 96c:03077
  • 20. W.A. Hodges, Model Theory, Cambridge University Press, 1993. MR 94e:03002
  • 21. S. Lang, Division points on curves, Ann. Mat. Pura Appl. LXX (1965), 229-234. MR 32:7560
  • 22. S. Lang, Number Theory III: Diophantine Geometry, Encyclopedia of Math. Sciences vol. 60, Springer-Verlag, 1991. MR 93a:11048
  • 23. A. J. Macintyre, K. McKenna and L. van den Dries, Elimination of quantifiers in algebraic structures, Advances in Mathematics 47 (1983), 74-87. MR 84f:03028
  • 24. Yu. Manin, Rational points of algebraic curves over function fields, AMS Translations Ser. II 59 (1966), 189-234.
  • 25. 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.
  • 26. 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.
  • 27. M. Messmer, Groups and fields interpretable in separably closed fields, Transactions of AMS 344 (1994), 361-377. MR 95c:03086
  • 28. A. Pillay, Geometric Stability Theory, Oxford University Press, 1996. CMP 97:07
  • 29. M. McQuillan, Division points on semi-abelian varieties, Inventiones Math. 120 (1995), 143-159. MR 96b:14020
  • 30. M. Raynaud, Sous-variétés d'une variété abélienne et points de torsion, in Arithmetic and Geometry, Vol I, Birkhauser, 1983. MR 85k:14022
  • 31. P. Samuel, Compléments à un article de Hans Grauert sur la conjecture de Mordell, Pbl. Math. IHES 29 (1966), 55-62. MR 34:4272
  • 32. I. R. Shafarevich, Basic Algebraic Geometry I, Springer-Verlag, 1994. MR 95m:14001
  • 33. J-P. Serre, Lectures on the Mordell-Weil Theorem, Vieweg, 1989. MR 90e:11086
  • 34. S. Shelah, Classification Theory, North-Holland, 1990. MR 91k:03085

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 03C60, 14G05

Retrieve articles in all journals with MSC (1991): 03C60, 14G05


Additional Information

Anand Pillay
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: pillay@math.uiuc.edu

DOI: https://doi.org/10.1090/S0273-0979-97-00730-1
Additional Notes: Partially supported by NSF grant DMS 96-96268.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society