Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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

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).


References [Enhancements On Off] (What's this?)

  • 1. D. Abramovich, Subvarieties of semiabelian varieties, Compositio Math, 90 (1994), 37-52.MR 1266493 (95c:14054)
  • 2. Z. Chatzidakis and E. Hrushovski, The model theory of difference fields, Transactions AMS 351 (1999), 2997-3071.MR 1652269 (2000f:03109)
  • 3. Z. Chatzidakis, E. Hrushovski, and Y. Peterzil, Model theory of difference fields II, Proceedings London Math. Soc. (3) 85 (2002), 257-311. MR 1912052 (2004c:03047)
  • 4. E. Hrushovski, Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690.MR 1333294 (97h:11154)
  • 5. E. Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Annals of Pure and Applied Logic 112 (2001), 43-115.MR 1854232 (2003d:03061)
  • 6. J. E. Humphreys, Linear Algebraic Groups, Springer, 1975.MR 0396773 (53:633)
  • 7. P. Kowalski and A. Pillay, Quantifier-elimination for algebraic $ D$-groups, Transactions AMS 358 (2006), 167-181.MR 2171228
  • 8. 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/.)
  • 9. A. Pillay, Mordell-Lang conjecture for function fields in characteristic zero, revisited, Compositio Math. 140 (2004), 64-68.MR 2004123 (2005b:14079)
  • 10. A. Pillay and M. Ziegler, Jet spaces of varieties over differential and difference fields, Selecta Math. New Ser. 9 (2003), 579-599.MR 2031753 (2004m:12011)
  • 11. R. Pink and D. Roessler, On Hrushovski's proof of the Manin-Mumford conjecture, Proceedings ICM 2002, Vol. I, Higher Education Press, 2002. MR 1989204 (2004f:14062)
  • 12. R. Pink and D. Roessler, On $ \psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture, Journal of Algebraic Geometry 13 (2004), 771-798.MR 2073195 (2005d:14061)
  • 13. T. Scanlon, Positive characteristic Manin-Mumford, Compositio Math. 141 (2005), 1351-1364.MR 2185637
  • 14. J.-P. Serre, Groupes algebriques et corps de classes, Hermann, 1956.MR 0103191 (21:1973)
  • 15. Andre Weil, On algebraic groups of transformations, American J. Math. 77 (1955).MR 0074083 (17:533e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14K12

Retrieve articles in all journals with MSC (2000): 14K12


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

American Mathematical Society