Skip to main content
Log in

Local points of twisted Mumford quotients and Shimura curves

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

We determine over which fields twisted Mumford quotients have rational points. Using the $p$-adic uniformization, we apply these results to Shimura curves, and show some new cases for which the jacobians are even in the sense of [PS].

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

Access this article

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. Berkovich, V.G.: Spectral Theory and Analytic Geometry over non-Archimedean Fields. Mathematical Surveys and Monographs 33, American Mathematical Society, Providence, 1990

  2. Borevich, A.I., Shafarevich, I.R.: Number Theory. Academic Press, New York- London, 1966

  3. Cherednik, I.V.: Uniformization of algebraic curves by discrete arithmetic subgroups of PGL 2(K w ) with compact quotients. Math. USSR Sb. 29, 55–85 (1976)

    MATH  Google Scholar 

  4. Chinburg, T., Friedman, E.: An embedding theorem for quaternion algebras. J. London Math. Soc. 60, 33–44 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deligne, P.: Variétés de Shimura : Interprétation modulaire, et techniques de construction de modèles canoniques. Proc. Symposia in Pure Math. 33, 247–290 (1979)

    MATH  Google Scholar 

  6. Deligne, P.: Travaux de Shimura , Sém. Bourbaki. Fév. 1971, Exposé 389, Lecture Notes Math. 244, Springer, Heidelberg, 1971

  7. Jordan, B.W., Livné, R.: Local diophantine properties of Shimura curves. Math. Ann. 270, 235–248 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Jordan, B.W., Livné, R.: Divisor classes on Shimura curves rational over local fields. J. reine angew. Math. 378, 46–52 (1985)

    MATH  Google Scholar 

  9. Jordan, B.W., Livné, R.: On Atkin-Lehner quotients of Shimura curves. Bull. London Math. Soc. 31, 681–685 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kurihara, A.: On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo, Sec. 1A 25, 277–301 (1979)

    Google Scholar 

  11. Milne, J.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: L. Clozel and J. S. Milne, Automorphic Forms, Shimura Varieties, and L-functions, Perspectives in Mathematics 10, Academic Press, San Diego, 1990

  12. Morita, Y.: Reduction modulo ᵊB of Shimura curves. Hokkaido Math. J. 10, 209–238 (1981)

    MathSciNet  MATH  Google Scholar 

  13. Mumford, D.: An analytic construction of degenerating curves over complete local rings. Compositio Math. 24, 129–174 (1972)

    MATH  Google Scholar 

  14. Mustafin, G.A.: Nonarchimedean uniformization. Math. USSR Sb. 34, 187–214 (1978)

    MATH  Google Scholar 

  15. Poonen, B., Stoll, M.: The Cassels-Tate pairing on polarized abelian varieties. Ann. Math. 150, 1109–1149 (1999)

    MathSciNet  MATH  Google Scholar 

  16. Sadykov, M.: Parity of genera of Shimura curves over a real quadratic field. Preprint 336 at the Algebraic Number Theory Archives, available at http://www.math.uiuc.edu/Algebraic-Number-Theory, 2/14/02

  17. Shimura, G.: On the real points of an arithmetic quotient of a bounded symmetric domain. Math. Ann. 215, 135–164 (1975)

    MATH  Google Scholar 

  18. Teitelbaum, J.: On Drinfeld’s universal formal group on the p -adic upper half plane. Math. Ann. 284, 647–674 (1989)

    MathSciNet  MATH  Google Scholar 

  19. Varshavsky, Y.: p-adic uniformization of unitary Shimura varieties. Publ. Math. IHES 87, 57–119 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Varshavsky, Y.: P-adic uniformization of unitary Shimura varieties, II. J. Diff. Geom. 49, 75–113 (1998)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bruce W. Jordan.

Additional information

Mathematics Subject Classification (2000): 14G20, 14G35

The first author was partially supported by grants from the NSF and PSC-CUNY

The first two authors were partially supported by a joint Binational Israel-USA Foundation grant

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jordan, B., Livné, R. & Varshavsky, Y. Local points of twisted Mumford quotients and Shimura curves. Math. Ann. 327, 409–428 (2003). https://doi.org/10.1007/s00208-003-0448-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-003-0448-3

Keywords

Navigation