The André-Oort conjecture for products of Drinfeld modular curves

Breuer, Florian

doi:10.1515/crll.2005.2005.579.115

Cited by 13 publications

(20 citation statements)

References 22 publications

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“…Yafaev has extended [10] to the case of arbitrary curves in Shimura varieties, assuming GRH (see [22]), and he has generalised a result of Moonen to arbitrary Shimura varieties in [23]. In the mean time, Breuer has succeeded in adapting the arguments of this article to the case of Drinfel'd modular curves in positive characteristic, see [2] and [3].…”

Section: Theoremmentioning

confidence: 91%

Special points on products of modular curves

Edixhoven¹

2005

Duke Math. J.

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We prove the André-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n ≥ 0, and let Σ be a subset of C n consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of Σ are special sub-varieties, i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant (Thm. 1.3) is proved unconditionally.

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Section: Theoremmentioning

confidence: 91%

Special points on products of modular curves

Edixhoven¹

2005

Duke Math. J.

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“…Results analogous to (A) above were obtained by the author for products of Drinfeld modular curves in odd characteristic [3,4]. Here special points are called CM points, as they correspond to tuples of rank 2 Drinfeld modules with complex multiplication.…”

Section: Characteristic Pmentioning

confidence: 72%

Special subvarieties of Drinfeld modular varieties

Breuer

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We explore an analogue of the André-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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“…It follows that T N is also described by the inclusion Y 0 (N ) ⊂ M 2 from (2.11). The correspondence T N therefore coincides with what we called T m in [3]. In the remainder of this section, we closely follow [3, §2].…”

Section: Points On Products Of Drinfeld Modular Curves 1363mentioning

confidence: 76%

“…In fact, our proof is closely modeled on Edixhoven's approach [4]. Theorem 1.2 was proved in [3] for the special case A = F q [T ]. In this paper we show how to adapt the arguments of [3] to the case of general A (but still of odd characteristic).…”

Section: Theorem 12 Let X = X 1 × · · · × X N Be a Product Of Drinfmentioning

confidence: 99%

“…We remark that the GL 2 (k ∞ ) action on Ω induces a PGL 2 (k ∞ ) action, as the center acts trivially. Until now we have found it more convenient to work with GL 2 , but in order to continue we will need a number of group-theoretic results, and here it will be simpler to work with PGL 2 , as had been done throughout [3]. For the convenience of the reader, we recall here some basic properties of the groups PGL 2 (k ∞ ), which we will need.…”

Section: Preimages Inmentioning

confidence: 99%

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