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 Free Access

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 Maninâ€™s homomorphism and of a certain analog in characteristic $p$.

- Dan Abramovich and JosĂ© Felipe Voloch,
*Toward a proof of the Mordell-Lang conjecture in characteristic $p$*, Internat. Math. Res. Notices**5**(1992), 103â€“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.Â 3, 557â€“567. MR**1189865**, DOI 10.2307/2946600 - Alexandru Buium,
*Effective bound for the geometric Lang conjecture*, Duke Math. J.**71**(1993), no.Â 2, 475â€“499. MR**1233446**, DOI 10.1215/S0012-7094-93-07120-7 - Alexandru Buium and JosĂ© Felipe Voloch,
*Integral points of abelian varieties over function fields of characteristic zero*, Math. Ann.**297**(1993), no.Â 2, 303â€“307. MR**1241808**, DOI 10.1007/BF01459503 - FranĂ§oise Delon,
*IdĂ©aux et types sur les corps sĂ©parablement clos*, MĂ©m. Soc. Math. France (N.S.)**33**(1988), 76 (French, with English summary). MR**986208** - Ehud Hrushovski,
*Unidimensional theories are superstable*, Ann. Pure Appl. Logic**50**(1990), no.Â 2, 117â€“138. MR**1081816**, DOI 10.1016/0168-0072(90)90046-5 - U. Hrushovski and A. Pillay,
*Weakly normal groups*, Logic colloquium â€™85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp.Â 233â€“244. MR**895647**, DOI 10.1016/S0049-237X(09)70556-7 - E. Hrushovski and Z. Sokolovic,
*Minimal subsets of differentially closed fields*, Trans. Amer. Math. Soc. (to appear). - E. Hrushovski and B. Zilâ€™ber,
*Zariski geometries*, J. Amer. Math. Soc.**9**(1996), 1â€“56. - Ehud Hrushovski and Boris Zilber,
*Zariski geometries*, Bull. Amer. Math. Soc. (N.S.)**28**(1993), no.Â 2, 315â€“323. MR**1183999**, DOI 10.1090/S0273-0979-1993-00380-X - Serge Lang,
*Division points on curves*, Ann. Mat. Pura Appl. (4)**70**(1965), 229â€“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 - Daniel Lascar,
*Ranks and definability in superstable theories*, Israel J. Math.**23**(1976), no.Â 1, 53â€“87. MR**409169**, DOI 10.1007/BF02757234 - Ju. I. Manin,
*Algebraic curves over fields with differentiation*, Izv. Akad. Nauk SSSR. Ser. Mat.**22**(1958), 737â€“756 (Russian). MR**0103889** - Ju. I. Manin,
*Rational points on algebraic curves over function fields*, Izv. Akad. Nauk SSSR Ser. Mat.**27**(1963), 1395â€“1440 (Russian). MR**0157971** - M. Messmer, Groups and fields interpretable in separably closed fields, preprint.
- 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. 1â€“22. - A. Robinson and P. Roquette,
*On the finiteness theorem of Siegel and Mahler concerning Diophantine equations*, J. Number Theory**7**(1975), 121â€“176. MR**374022**, DOI 10.1016/0022-314X(75)90013-X - Gerald E. Sacks,
*Saturated model theory*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR**0398817** - Z. Sokolovic,
*Model theory of differential fields*, Ph.D. Thesis, Notre Dame, July, 1992. - Morgan Ward,
*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783â€“787. MR**10**, DOI 10.2307/2371336 - Carol Wood,
*Notes on the stability of separably closed fields*, J. Symbolic Logic**44**(1979), no.Â 3, 412â€“416. MR**540671**, DOI 10.2307/2273133

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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

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