Manin kernels (Q2758393)

In his proof of the Mordell conjecture for function fields, \textit{Yu. Manin} [Am. Math. Soc., Transl. II. Ser. 50, 189--234 (1966; Zbl 0178.55102)] showed that if \(A\) is an abelian variety defined over a differential field, then here is a homomorphism \(\mu\) of differential algebraic groups from \(A\) into a vector group. The kernel \(A^\#\) of \(\mu\) plays an important role in both \textit{A. Buium's} [``Differential Algebra and Diophantine Geometry'' (1994; Zbl 0870.12007)] and \textit{E. Hrushovski's} [J. Am. Math. Soc. 9, 667--690 (1996; Zbl 0864.03026)] proofs of the Mordell-Lang conjecture for characteristic zero function fields. My goal in this note is to give an exposition of one construction of the Manin kernel using the notion of prolongations of algebraic varieties. The treatment of prolongations that I give here takes a rather naive point of view compared to the scheme theoretic treatment in \textit{A. Buium's} paper [loc. cit.]. For smooth varieties the two treatments agree, while for general varieties our prolongations correspond to the reduced points of those of \textit{A. Buium's} paper [loc. cit.]. Section \S3 discusses Kolchin's logarithmic-derivative map for algebraic groups defined over the constant field. Though in the spirit of the main theme, this material in not necessary for later sections.NEWLINENEWLINE Manin kernels are of great model theoretic interest. If \(A\) is a simple abelian variety that does not descend to the constants, then \textit{E. Hrushovski} and \textit{Z. Sokolovic} [``Minimal subsets of differentially closed fields'', to appear in Trans. Am. Math. Soc.] showed that \(A^\#\) is a locally modular strongly animal set. In \S5 this proof is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00010].