Integral points of abelian varieties over function fields of characteristic zero
The main result of the paper asserts that if \(K\) is a function field of characteristic zero and \(v\) is a place of \(K\) then the local height \(\lambda_ v(X,P)\) associated to a closed subvariety \(X\) of an abelian \(K\)-variety \(A\) with trace zero is bounded for \(P \in A(K) \backslash X\). This is a higher dimensional version of a result of \textit{Yu. I. Manin} on elliptic curves [Transl., Ser. 2, Am. Math. Soc. 50, 189-234 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)]. If in addition \(X\) is an ample divisor, then one obtains as a corollary that for any finite set of places \(S\) of \(K\) the set of \(S\)-integral points of \(A \backslash X\) is finite; this is the function field analogue of a conjecture of Lang proved by \textit{G. Faltings} [Ann. Math., II. Ser. 133, No. 3, 549-576 (1991; Zbl 0734.14007)] and extends a previous result of \textit{A. N. Parshin} [The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 163-178 (1990; Zbl 0742.14018)].
scientific article; zbMATH DE number 537350