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Corrigendum: Torsion points on abelian varieties of CM-type

Part of: Arithmetic problems. Diophantine geometry Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 December 2017

A. Silverberg
A. Silverberg*
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697-3875, USA email asilverb@uci.edu
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Abstract

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Keywords

abelian varietiescomplex multiplicationtorsion pointsrational points

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 11G15: Complex multiplication and moduli of abelian varieties 14G05: Rational points 14K22: Complex multiplication
Type
Corrigendum
Information
Compositio Mathematica , Volume 154 , Issue 3 , March 2018 , pp. 594
Copyright
© The Author 2017 

As pointed out in [Reference Gaudron and RémondGR17, § 3], Lemma 6 of [Reference SilverbergSil88] is false as stated. Thus, the proofs of the results that rely on it are not correct as stated. These include the proofs of [Reference SilverbergSil88, Corollaries 4–8], the proofs of [Reference SilverbergSil92, Corollaries 5.2(i), 5.3(i), and 5.4(i,ii)], and the proofs of the results in [Reference van MulbregtvMu92] for non-simple two- and three-dimensional abelian varieties. The proof given in [Reference SilverbergSil88, Lemma 6] is valid with the additional assumption that the abelian variety $A$ is (absolutely) simple. Even without that assumption, the proofs of the corollaries can be salvaged in certain cases; however, stronger results have recently been achieved in [Reference Gaudron and RémondGR17].

I thank Gaël Rémond and Éric Gaudron for pointing out this error.

References

Gaudron, É. and Rémond, G., Torsion des variétés abéliennes CM, Proc. Amer. Math. Soc., to appear, doi:10.1090/proc/13885. Preprint (2017), http://math.univ-bpclermont.fr/∼gaudron/art17.pdf.Google Scholar
Silverberg, A., Torsion points on abelian varieties of CM-type , Compos. Math. 68 (1988), 241–249; http://www.numdam.org/item?id=CM_1988__68_3_241_0.Google Scholar
Silverberg, A., Points of finite order on abelian varieties , in p-adic methods in number theory and algebraic geometry, Contemporary Mathematics, vol. 133 (American Mathematical Society, Providence, RI, 1992), 175–193.Google Scholar
van Mulbregt, P., Torsion-points on low dimensional abelian varieties with complex multiplication , in p-adic methods in number theory and algebraic geometry, Contemporary Mathematics, vol. 133 (American Mathematical Society, Providence, RI, 1992), 205–210.Google Scholar