On algebraic 𝜎-groups

Kowalski, Piotr; Pillay, Anand

doi:10.1090/S0002-9947-06-04312-1

On algebraic $\sigma$ -groups

Authors: Piotr Kowalski and Anand Pillay
Journal: Trans. Amer. Math. Soc. 359 (2007), 1325-1337
MSC (2000): Primary 14K12
DOI: https://doi.org/10.1090/S0002-9947-06-04312-1
Published electronically: October 17, 2006
MathSciNet review: 2262852
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Abstract: We introduce the categories of algebraic $\sigma$ -varieties and $\sigma$ -groups over a difference field $(K,\sigma)$ . Under a ``linearly $\sigma$ -closed" assumption on $(K,\sigma)$ we prove an isotriviality theorem for $\sigma$ -groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski's work (2001) and are also closely related to papers of Pink and Roessler (2002 and 2004).

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Additional Information

Piotr Kowalski
Affiliation: Department of Mathematics, University of Wroclaw, pl Grunwaldzki 2/4, 50-384 Wroclaw, Poland – and – Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2975

Anand Pillay
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2975 – and – School of Mathematics, University of Leeds, Leeds, England LS2 9JT

DOI: https://doi.org/10.1090/S0002-9947-06-04312-1
Received by editor(s): January 28, 2005
Published electronically: October 17, 2006
Additional Notes: The first author was supported by funds from NSF Focused Research Grant DMS 01-00979, and by the Polish KBN grant 2 P03A 018 24
The second author was supported by NSF grants
Article copyright: © Copyright 2006 American Mathematical Society