[PDF][PDF] Diophantine geometry of the torsion of a Drinfeld module

T Scanlon - Journal of number theory, 2002 - core.ac.uk
Journal of number theory, 2002core.ac.uk
DENIS proposed that the qualitative diophantine results known for certain subgroups of semi-
abelian varieties should hold for Drinfeld modules. In particular, the Manin–Mumford
conjecture which asserts that an irreducible subvariety of a semi-abelian variety containing a
Zariski dense set of torsion points must itself be a translate of a subalgebraic group should
be true with ''semi-abelian variety''replaced by ''power of the additive group considered as an
Fp½tŠ-module via a Drinfeld module.''In [4], Denis permits finite extensions of Fp½tŠ but …
DENIS proposed that the qualitative diophantine results known for certain subgroups of semi-abelian varieties should hold for Drinfeld modules. In particular, the Manin–Mumford conjecture which asserts that an irreducible subvariety of a semi-abelian variety containing a Zariski dense set of torsion points must itself be a translate of a subalgebraic group should be true with ‘‘semi-abelian variety’’replaced by ‘‘power of the additive group considered as an Fp½tŠ-module via a Drinfeld module.’’In [4], Denis permits finite extensions of Fp½tŠ but insists that the Drinfeld module have generic characteristic. The strengthening permits one to consider general Drinfeld modules while the restriction is necessary since every point of GağFalg p Ş is a torsion point for every Drinfeld module of finite characteristic. The analogue of Boxall’s theorem [1] may still be true for
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