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By Mr (i: G (k and Wintenberger J. -p. (f-stras


Démonstration d’une conjecture de Lang dans des cas particuliers. (French. English summary) [Proof of a conjecture by Lang in some particular cases] J. Reine Angew. Math. 553 (2002), 1–16. In the early 1960s, Mumford asked the following question: If a complex curve in its Jacobian contains infinitely many torsion points, is it necessarily of genus 1? Lang observed that the answer is yes under the following hypothesis: let A be an abelian variety over a field K finitely generated over Q; then there exists an integer c ≥ 1 such that, for every n and point x of order n on A, the subgroup of (Z/nZ) × consisting of the classes [d] such that dx is conjugate to x over K has index ≤ c in the full group. For an abelian variety A over a number field K, this amounts to the following pair of conditions on the l-adic representations ρl: GK → Aut(TlA) attached to A: (a) for all l, ρl(GK) contains an open subgroup of the group Z × l of homotheties; (b) for all but finitely many l, ρl(GK) contains Z × l. Bogomolov proved (a) and Serre showed that, for a fixed A/K, the indices (Z × l: Z

Year: 2013
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