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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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).
- Dan Abramovich, Subvarieties of semiabelian varieties, Compositio Math. 90 (1994), no. 1, 37–52. MR 1266493
- Zoé Chatzidakis and Ehud Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc. 351 (1999), no. 8, 2997–3071. MR 1652269, DOI https://doi.org/10.1090/S0002-9947-99-02498-8
- Zoé Chatzidakis, Ehud Hrushovski, and Ya’acov Peterzil, Model theory of difference fields. II. Periodic ideals and the trichotomy in all characteristics, Proc. London Math. Soc. (3) 85 (2002), no. 2, 257–311. MR 1912052, DOI https://doi.org/10.1112/S0024611502013576
- Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR 1333294, DOI https://doi.org/10.1090/S0894-0347-96-00202-0
- Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115. MR 1854232, DOI https://doi.org/10.1016/S0168-0072%2801%2900096-3
- James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
- Piotr Kowalski and Anand Pillay, Quantifier elimination for algebraic $D$-groups, Trans. Amer. Math. Soc. 358 (2006), no. 1, 167–181. MR 2171228, DOI https://doi.org/10.1090/S0002-9947-05-03820-1
- J. Oesterlé, La conjecture de Manin-Mumford (d’apres Pink et Roessler), electronic letter to D. Roessler, 20 Dec. 2002. (See preprints at http://www.math.ethz.ch/ roessler/.)
- Anand Pillay, Mordell-Lang conjecture for function fields in characteristic zero, revisited, Compos. Math. 140 (2004), no. 1, 64–68. MR 2004123, DOI https://doi.org/10.1112/S0010437X03000186
- Anand Pillay and Martin Ziegler, Jet spaces of varieties over differential and difference fields, Selecta Math. (N.S.) 9 (2003), no. 4, 579–599. MR 2031753, DOI https://doi.org/10.1007/s00029-003-0339-1
- Richard Pink and Damian Roessler, On Hrushovski’s proof of the Manin-Mumford conjecture, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 539–546. MR 1989204
- Richard Pink and Damian Roessler, On $\psi $-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture, J. Algebraic Geom. 13 (2004), no. 4, 771–798. MR 2073195, DOI https://doi.org/10.1090/S1056-3911-04-00368-6
- Thomas Scanlon, A positive characteristic Manin-Mumford theorem, Compos. Math. 141 (2005), no. 6, 1351–1364. MR 2185637, DOI https://doi.org/10.1112/S0010437X05001879
- Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’institut de mathématique de l’université de Nancago, VII. Hermann, Paris, 1959 (French). MR 0103191
- André Weil, On algebraic groups of transformations, Amer. J. Math. 77 (1955), 355–391. MR 74083, DOI https://doi.org/10.2307/2372535
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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
Article copyright:
© Copyright 2006
American Mathematical Society
