## On algebraic $\sigma$-groups

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- by Piotr Kowalski and Anand Pillay PDF
- Trans. Amer. Math. Soc.
**359**(2007), 1325-1337 Request permission

## 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).## References

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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
- MR Author ID: 658570
**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
- MR Author ID: 139610
- 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 - © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**359**(2007), 1325-1337 - MSC (2000): Primary 14K12
- DOI: https://doi.org/10.1090/S0002-9947-06-04312-1
- MathSciNet review: 2262852