On algebraic 𝜎-groups

Kowalski, Piotr; Pillay, Anand

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

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).

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