Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

The Mordell-Lang conjecture
for function fields


Author: Ehud Hrushovski
Journal: J. Amer. Math. Soc. 9 (1996), 667-690
MSC (1991): Primary 03C45, 11G10; Secondary 03C60, 14G05, 12H05
DOI: https://doi.org/10.1090/S0894-0347-96-00202-0
MathSciNet review: 1333294
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin's homomorphism and of a certain analog in characteristic $p$.


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

  • [AV91] D. Abramovich and J. F. Voloch Toward a proof of the Mordell-Lang conjecture in characteristic $p$, Internat. Math. Res. Notices 5 (1992), 103--115. MR 94f:11051
  • [Bu92] A. Buium, Intersections in jet spaces and a conjecture of S. Lang, Ann. of Math. (2) 136 (1992), 557--567.MR 93j:14055
  • [Bu93] ------, Effective bound for the geometric Lang conjecture, Duke Math. J. 71 (1993), 475--499.MR 95c:14055
  • [BV93] A. Buium and J. F. Voloch, Integral points of abelian varieties over function fields of characteristic zero, Math. Ann. 297 (1993), 303--307.MR 94i:14029
  • [Del88] F. Delon, Idéaux et types sur les corps séparablement clos, Supplément au Mém. Soc. Math. France (N.S.), vol. 116, no. 33, Soc. Math. France, Paris, 1988.MR 90m:03067
  • [Hr90] E. Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), 117--138.MR 92g:03052
  • [HP86] E. Hrushovski and A. Pillay, Weakly normal groups, Logic Colloquium '85, North-Holland, Amsterdam, 1987.MR 88e:03051
  • [HS] E. Hrushovski and Z. Sokolovic, Minimal subsets of differentially closed fields, Trans. Amer. Math. Soc. (to appear).
  • [HZ] E. Hrushovski and B. Zil'ber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), 1--56. CMP 95:06
  • [HZ93] ------, Zariski geometries, Bull. Amer. Math. Soc. (N.S.) 28 (1993), 315--323.MR 93j:14003
  • [La65] S. Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229--234.MR 32:7560
  • [La91] ------, Number Theory III: Diophantine geometry, Encyclopaedia Math. Sci., vol. 60, Springer-Verlag, Berlin, Heidelberg, and New York, 1991.MR 93a:11048
  • [Las] D. Lascar, Rank and definability in superstable theories, Israel J. Math. 23 (1976), 53--87.MR 53:12931
  • [Ma58] Yu. Manin, Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 737--756; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 37, Amer. Math. Soc., Providence, RI, 1964, pp. 59--78.MR 21:2652
  • [Ma63] ------, Rational points of algebraic curves over function fields, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1395--1440; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 59, Amer. Math. Soc., Providence, RI, 1966, pp. 189--234.MR 28:1199
  • [Mes] M. Messmer, Groups and fields interpretable in separably closed fields, preprint.
  • [NeP89] A. Pillay, Model theory, stability theory, and stable groups, The Model Theory of Groups (A. Nesin and A. Pillay, eds.), Notre Dame Math. Lectures, no. 11, Univ. Notre Dame Press, Notre Dame, IN, 1989, pp. 1--22.CMP 21:09
  • [RR75] A. Robinson and P. Roquette, On the finiteness theorem of Siegel and Mahler concerning diophantine equations, J. Number Theory 7 (1975), 121--176.MR 51:10222
  • [Sa72] G. Sacks, Saturated model theory, W. A. Benjamin, Reading, MA, 1972. MR 53:2668
  • [So92] Z. Sokolovic, Model theory of differential fields, Ph.D. Thesis, Notre Dame, July, 1992.
  • [Weil48] A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948.MR 10:621d
  • [Wood79] C. Wood, Notes on the stability of separably closed fields, J. Symbolic Logic 44 (1979), 412--416.MR 81m:03042

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 03C45, 11G10, 03C60, 14G05, 12H05

Retrieve articles in all journals with MSC (1991): 03C45, 11G10, 03C60, 14G05, 12H05


Additional Information

Ehud Hrushovski
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-277, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Hebrew University, Jerusalem, Israel
Email: ehud@math.mit.edu

DOI: https://doi.org/10.1090/S0894-0347-96-00202-0
Received by editor(s): September 1, 1993
Received by editor(s) in revised form: November 1, 1994
Additional Notes: The author was supported by the National Science Foundation
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society