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Abstract

Abstract. We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C 2 , containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m 1 such that X is the image, under the usual map, of the modular curve Y0m. The proof uses some number theory and some topological arguments.Mathematics Subject Classifications (1991): 11F32, 11G15, 14K22, 14G35.

O-minimality and the André-Oort conjecture for C^n

“…Note that in this case the results of Edixhoven [34] are unconditional and surely effective as well. Under GRH, the uniformity in the conclusion for curves of fixed degree was observed in [32], and this was shown to be effective and extended to curves in C n by Breuer [19].…”

“…As already mentioned, André [3] proved AO unconditionally for a product of two modular curves. Independently, Edixhoven [32] proved the same under GRH for imaginary quadratic fields, and later, under the same GRH assumptions, for an arbitrary product of modular curves [34] (see also [92]). Under GRH for suitable CM fields, Yafaev [95] affirms AO for products of two Shimura curves, Edixhoven [33] for Hilbert modular surfaces, and Yafaev [96] for curves in an arbitrary Shimura variety.…”

“…Under the Generalized Riemann Hypothesis for imaginary quadratic fields this result is due to Edixhoven [32], [34]; for n = 2 it is an unconditional result of André [3]. Our approach uses the theory of o-minimal structures, a part of Model Theory.…”

An effective result of André-Oort type

“…Under the Generalized Riemann Hypothesis this conjecture was proved by Edixhoven and Yafaev for certain simple Shimura varieties Sh K (G, X ) [8], [9], [10], [28] and eventually by Klingler, Ullmo, and Yafaev [18], [26] in general. Edixhoven and Yafaev also proved the above conjecture unconditionally if all points of S are contained in one Hecke orbit [11].…”

A mod p variant of the André–Oort conjecture

“…In the simplest non-trivial case of this conjecture the Shimura variety S is C 2 , the product of two copies of the j-line, hence the coarse moduli space for pairs of complex elliptic curves. The irreducible special curves in C 2 are, apart from the fibres of the two projections, the images of the modular curves Y 0 (n) (n ≥ 1), and consist of the pairs ( j(E), j(E/ P )) with E a complex elliptic curve and P ∈ E of order n. In this case, the conjecture was proved in [1], and, conditionally on the generalised Riemann hypothesis (GRH) for quadratic fields, in [4]. In this article we state a variant in positive characteristic, and prove it under GRH for quadratic fields.…”