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Diophantine geometry in characteristic $$p$$: A survey. (English) Zbl 0905.14011
Catanese, Fabrizio (ed.), Arithmetic geometry. Proceedings of a symposium, Cortona, Arezzo, Italy, October 16–21, 1994. Cambridge: Cambridge University Press. Symp. Math. 37, 260-278 (1997).
As its title suggests, this paper is a survey of diophantine geometry in positive characteristic. After a brief historical introduction, it describes results on “effective Mordell” (for bounding the heights of rational points on curves over function fields of characteristic $$p>0$$) by M. Kim [Compos. Math. 105, No. 1, 43-54 (1997; Zbl 0871.14020)]; bounds on the number of rational points on such curves by A. Buium and J. F. Voloch [Compos. Math. 103, No. 1, 1-6 (1996; Zbl 0885.14010)]; the analogue in characteristic $$p>0$$ of a conjecture of S. Lang on closed subvarieties of semiabelian varieties by E. Hrushovski [J. Am. Math. Soc. 9, No. 3, 667-690 (1996; Zbl 0864.03026)]; and work on an analogue of Roth’s theorem at one valation $$v$$ by (among others) A. Lasjaunias and B. de Mathan [J. Reine Angew. Math. 473, 195-206 (1996; Zbl 0844.11046)].
The paper concludes with an appendix that discusses various types of isotriviality that can occur in characteristic $$p>0$$.
For the entire collection see [Zbl 0864.00054].
Reviewer: P.Vojta (Berkeley)

MSC:
 14G05 Rational points 14G20 Local ground fields in algebraic geometry 11G99 Arithmetic algebraic geometry (Diophantine geometry) 14H25 Arithmetic ground fields for curves