Skip to main content
SpringerLink
Log in
Book cover

Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds pp 143–152Cite as

Explicit Algebraic Coverings of a Pointed Torus

  • Chapter
  • First Online:

  • 1633 Accesses

Part of the book series: Fields Institute Communications ((FIC,volume 67))

Abstract

This note contains an application of the algebraic study by Schütt and Shioda of the elliptic modular surface attached to the commutator subgroup of the modular group. This is used here to provide algebraic descriptions of certain coverings of a j-invariant 0 elliptic curve, unramified except over precisely one point.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. A.S.I. Anema, Branched covering spaces of an elliptic curve that branch only above a single point, Master’s thesis, Groningen, (2011) http://irs.ub.rug.nl/dbi/4e707a67dac82. Accessed 20 July 2012

  2. J.-I. Igusa, Fibre systems of Jacobian varieties: (III. Fibre systems of elliptic curves). Am. J. Math. 81, 453–476 (1959)

    Google Scholar 

  3. H.W. Lenstra, Galois theory for schemes. Lecture Notes (2008) http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf. Accessed on 20 July (2012)

  4. J. Maciel, SL n (F) equals its own derived group. Int. J. Algebra 2, 585–594 (2008)

    MathSciNet  MATH  Google Scholar 

  5. D.E. Rohrlich, Modular curves, Hecke correspondences, and L-functions, in Modular Forms and Fermat’s Last Theorem, ed. by G.Cornell, J.H. Silverman, G. Stevens (Springer, New York, 1997), pp. 41–100

    Chapter  Google Scholar 

  6. M. Schütt, T. Shioda, An interesting elliptic surface over an elliptic curve. Proc. Jpn. Acad. Sci. 83, 40–45 (2007)

    Article  MATH  Google Scholar 

  7. J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259–331 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Sijsling, Arithmetic (1; e)-curves and Belyĭ maps. Math. Comp. 81, 1823–1855 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Sijsling, Canonical models of arithmetic (1; e)-curves. Math. Zeitschrift 271, 38 (2012)

    Google Scholar 

  10. J.H. Silverman, The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106 (Springer, New York, 1986)

    Google Scholar 

  11. J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151 (Springer, New York, 1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. JBI-RuG, Nijenborgh 9, 9747 AG, Groningen, The Netherlands

    Ane S. I. Anema & Jaap Top

Authors
  1. Ane S. I. Anema
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaap Top
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaap Top .

Editor information

Editors and Affiliations

  1. , Mathematics Department, Stony Brook University, N/A, Stony Brook, 11794-3651, New York, USA

    Radu Laza

  2. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Matthias Schütt

  3. , Department of Math & Stats, Queen's University, University Ave., Kingston, K7L 3N6, Ontario, Canada

    Noriko Yui

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Anema, A.S.I., Top, J. (2013). Explicit Algebraic Coverings of a Pointed Torus. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation