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15 February 2005 Special points on products of modular curves
Bas Edixhoven
Duke Math. J. 126(2): 325-348 (15 February 2005). DOI: 10.1215/S0012-7094-04-12624-7

Abstract

We prove the André-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the generalized Riemann hypothesis (GRH). More explicitly, this means the following. Let n ≥ 0, and let Σ be a subset of ℂn 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 subvarieties, that is, are determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant (Th. 1.3) is proved unconditionally.

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Bas Edixhoven. "Special points on products of modular curves." Duke Math. J. 126 (2) 325 - 348, 15 February 2005. https://doi.org/10.1215/S0012-7094-04-12624-7

Information

Published: 15 February 2005
First available in Project Euclid: 21 January 2005

zbMATH: 1072.14027
MathSciNet: MR2110630
Digital Object Identifier: 10.1215/S0012-7094-04-12624-7

Rights: Copyright © 2005 Duke University Press

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Vol.126 • No. 2 • 15 February 2005
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