Special subvarieties of Drinfeld modular varieties

Details Special subvarieties of Drinfeld modular varieties Special subvarieties of Drinfeld modular varieties

2005-03-22

arxiv.org/abs/math/0503452v3

Mathematics

Number Theory

22 pages, significant rewrite

Scientific paper

We explore an analogue of the Andr\'e-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when $X$ contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when $X$ is a curve containing infinitely many CM points without any additional assumptions.

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