Skip to main content
SpringerLink
Log in
Book cover

2017 MATRIX Annals pp 487–489Cite as

Special Values of Hypergeometric Functions and Periods of CM Elliptic Curves

  • Chapter
  • First Online:

  • 1133 Accesses

Part of the book series: MATRIX Book Series ((MXBS,volume 2))

Abstract

Let \(X=X_0^6(1)/W_6\) be the quotient of the Shimura curve \(X_0^6(1)\) by all the Atkin-Lehner involutions. By realizing modular forms on X in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer’s formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over \(\overline Q\) with complex multiplication.

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   119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   159.99
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. Elkies, N.D.: Shimura curve computations via K3 surfaces of Néron-Severi rank at least 19. In: Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 5011, pp. 196–211. Springer, Berlin (2008)

    Google Scholar 

  2. Errthum, E.: Singular moduli of Shimura curves. Can. J. Math. 63(4), 826–861 (2011)

    Article  MathSciNet  Google Scholar 

  3. Guo, J.W., Yang, Y.: Equations of hyperelliptic Shimura curves. Compos. Math. 153, 1–40 (2017)

    Article  MathSciNet  Google Scholar 

  4. Nelson, P.D.: Evaluating modular forms on Shimura curves. Math. Comput. 84(295), 2471–2503 (2015)

    Article  MathSciNet  Google Scholar 

  5. Schofer, J.: Borcherds forms and generalizations of singular moduli. J. Reine Angew. Math. 629, 1–36 (2009)

    Article  MathSciNet  Google Scholar 

  6. Shimura, G.: Automorphic forms and the periods of abelian varieties. J. Math. Soc. Jpn. 31(3), 561–592 (1979)

    Article  MathSciNet  Google Scholar 

  7. Voight, J., Willis, J.: Computing power series expansions of modular forms. In: Computations with Modular Forms. Contrib. Math. Comput. Sci. vol. 6, pp. 331–361. Springer, Cham (2014)

    Google Scholar 

  8. Watson, T.C.: Rankin triple products and quantum chaos. ProQuest LLC, Ann Arbor, MI (2002). Thesis (Ph.D.), Princeton University

    Google Scholar 

  9. Yang, Y.: Schwarzian differential equations and Hecke eigenforms on Shimura curves. Compos. Math. 149(1), 1–31 (2013)

    Article  MathSciNet  Google Scholar 

  10. Yang, Y.: Ramanujan-type identities for Shimura curves. Isr. J. Math. 214(2), 699–731 (2016)

    Article  MathSciNet  Google Scholar 

  11. Yang, Y.: Special values of hypergeometric functions and periods of CM elliptic curves. Trans. Am. Math. Soc. 370, 6433–6467 (2018)

    Article  MathSciNet  Google Scholar 

  12. Yoshida, H.: Absolute CM-periods. Mathematical Surveys and Monographs, vol. 106. American Mathematical Society, Providence (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

    Yifan Yang

  2. National Center for Theoretical Sciences, Hsinchu, Taiwan

    Yifan Yang

Authors
  1. Yifan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yifan Yang .

Editor information

Editors and Affiliations

  1. The University of Melbourne, Melbourne, Australia

    Jan de Gier

  2. University of Western Australia, Perth, Australia

    Cheryl E. Praeger

  3. UCLA, Los Angeles, CA, USA

    Terence Tao

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Yang, Y. (2019). Special Values of Hypergeometric Functions and Periods of CM Elliptic Curves. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_42

Download citation

Publish with us

Policies and ethics

Search

Navigation