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A note on the Manin-Mumford conjecture. (English) Zbl 1098.14030
van der Geer, Gerard (ed.) et al., Number fields and function fields – two parallel worlds. Boston, MA: Birkhäuser (ISBN 0-8176-4397-4/hbk; 0-8176-4447-4/e-book). Prog. Math. 239, 311-318 (2005).
Summary: R. Pink and the author [in: Proc. ICM 2002, Beijing, China, Vol. I, 539–546 (2002; Zbl 1026.14012)] gave a short proof of the Manin-Mumford conjecture, which was inspired by an earlier model-theoretic proof by Hrushovski. The proof given in [loc. cit.] uses a difficult unpublished ramification-theoretic result of Serre. It is the purpose of this note to show how the proof given in [loc. cit.] can be modified so as to circumvent the reference to Serre’s result. J. Oesterle and R. Pink contributed several simplifications and shortcuts to this note.
MSC:
14K05Algebraic theory of abelian varieties
14K15Arithmetic ground fields (abelian varieties)