Skip to main content
SpringerLink
Log in

Local André-Oort conjecture for the universal abelian variety

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We prove a p-adic version of the André-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n≥3 and p a prime number not dividing n. Let R be a finite extension of \(W[\mathbb{F}_{p}^{alg}]\), the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space \(\mathcal{A}=\mathcal{A}_{g,1,n}\) of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety \(\pi:\mathcal{X}\to\mathcal{A}\) over \(\mathcal{A}\) may be defined over R. We call a point \(\xi\in\mathcal{X}(R)\) R-special if \(\mathcal{X}_{\pi(\xi)}\) is a canonical lift and ξ is a torsion point of its fibre. Employing the model theory of difference fields and work of Moonen on special subvarieties of \(\mathcal{A}\), we show that an irreducible subvariety of \(\mathcal{X}_{R}\) containing a dense set of R-special points must be a special subvariety in the sense of mixed Shimura varieties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. André, Y.: G-Functions and geometry. Aspects of Mathematics, vol. E13, xii+229 pp. Braunschweig: Friedr. Vieweg & Sohn 1989

  2. André, Y.: Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire. J. Reine Angew. Math. 505, 203–208 (1998)

    MATH  MathSciNet  Google Scholar 

  3. Bélair, L., Macintyre, A., Scanlon, T.: Model theory of Frobenius on Witt vectors. Preprint 2002, available at http://www.math.berkeley.edu/∼scanlon/papers/papers.html

  4. Chatzidakis, Z., Hrushovski, E.: Model theory of difference fields. Trans. Am. Math. Soc. 351, 2997–3071 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Clark, P.: Bounds for torsion on abelian varieties with integral moduli. Preprint 2004, arXiv:math.NT/0407264

  6. Cohn, R.M.: Difference algebra, xiv+355 pp. New York, London, Sydney: Interscience Publishers John Wiley & Sons 1965

  7. Edixhoven, B.: Special points on the product of two modular curves. Compos. Math. 114, 315–328 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Edixhoven, B.: On the André-Oort conjecture for Hilbert modular surfaces. Moduli of abelian varieties (Texel Island, 1999). Prog. Math., vol. 195, pp. 133–155. Basel: Birkhäuser 2001

  9. Edixhoven, B., Yafaev, A.: Subvarieties of Shimura varieties. Ann. Math. (2) 157, 621–645 (2003)

    Google Scholar 

  10. Hrushovski, E.: Proof of Manin’s theorem by reduction to positive characteristic. In: Model Theory and Algebraic Geometry: An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture, ed. by E. Bouscaren. Lect. Notes Math., vol. 1696, pp. 197–205. Berlin: Springer 1998

  11. Hrushovski, E.: The Manin-Mumford conjecture and the model theory of difference fields. Ann. Pure Appl. Logic 112, 43–115 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mattuck, A.: Abelian varieties over p-adic ground fields. Ann. Math. (2) 62, 92–119 (1955)

    Google Scholar 

  13. Messing, W.: The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes. Lect. Notes Math., vol. 264. Berlin: Springer 1972

  14. Milne, J.S.: Abelian varieties. In: Arithmetic geometry, ed. by G. Cornell and J.H. Silverman, pp. 103–150. New York: Springer 1986

  15. Moonen, B.: Linearity properties of Shimura varieties, part II. Compos. Math. 114, 3–35 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, third edition. Ergeb. Math. Grenzgeb., vol. 34. New York: Spinger 1994

  17. Oort, F.: Canonical liftings and dense sets of CM-points. In: Arithmetic geometry (Cortona, 1994), pp. 228–234. Sympos. Math., XXXVII. Cambridge: Cambridge Univ. Press 1997

  18. Pink, R.: Arithmetical Compactifications of Mixed Shimura Varieties. Ph.D. thesis, Friedrich-Wilhelms-Universität Bonn. Math. Schr., vol. 209, 1989

  19. Pink, R.: A Combination of the conjectures of Mordell-Lang and André-Oort. In: Geometric Methods in Algebra and Number Theory, ed. by F. Bogomolov and Y. Tschinkel. Prog. Math., vol. 253, pp. 251–282. Birkhäuser 2005

  20. Pink, R.: A Common Generalization of the Conjectures of André-Oort, Manin-Mumford, and Mordell-Lang. Preprint, April 2005, available at http://www.math.ethz.ch/∼pink/preprints.html

  21. Pink, R., Roessler, D.: On Hrushovski’s proof of the Manin-Mumford conjecture. Proceedings of the International Congress of Mathematicians, vol. I (Beijing, 2002), pp. 539–546. Beijing: Higher Ed. Press 2002

  22. Scanlon, T.: p-adic distance from torsion points of semi-abelian varieties. J. Reine Angew. Math. 499, 225–236 (1998)

    MATH  MathSciNet  Google Scholar 

  23. Scanlon, T.: The conjecture of Tate and Voloch on p-adic proximity to torsion. Int. Math. Res. Not., no. 17, 909–914 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Scanlon, T.: Quantifier elimination for the relative Frobenius. In: Valuation Theory and Its Applications Volume II, Conference Proceedings of the International Conference on Valuation Theory (Saskatoon, 1999), ed. by F.-V. Kuhlmann, S. Kuhlmann, and M. Marshall. Fields Institute Communications Series, pp. 323–352. Providence: Am. Math. Soc. 2003

  25. Scanlon, T.: Automatic uniformity. Int. Math. Res. Not., no. 62, 3317–3326 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, Evans Hall, Berkeley, CA, 94720-3480, USA

    Thomas Scanlon

Authors
  1. Thomas Scanlon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Scanlon.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Scanlon, T. Local André-Oort conjecture for the universal abelian variety. Invent. math. 163, 191–211 (2006). https://doi.org/10.1007/s00222-005-0460-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-005-0460-1

Keywords

Search

Navigation