Skip to main content

Special values of anticyclotomic L-functions modulo λ


The purpose of this article is to generalize some results of Vatsal on the special values of Rankin–Selberg L-functions in an anticyclotomic \({\mathbb{Z}_{p}}\)-extension. Let g be a cuspidal Hilbert modular newform of parallel weight \({(2,\ldots,2)}\) and level \({\mathcal{N}}\) over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant \({\mathcal{D}}\). We study the l-adic valuation of the special values \({L(g,\chi,\frac{1}{2})}\) as \({\chi}\) varies over the ring class characters of K of \({\mathcal{P}}\)-power conductor, for some fixed prime ideal \({\mathcal{P}}\) . We prove our results under the only assumption that the prime to \({\mathcal{P}}\) part of \({\mathcal{N}}\) is relatively prime to \({\mathcal{D}}\).

This is a preview of subscription content, access via your institution.


  1. Bertolini M., Darmon H.: A rigid analytic Gross–Zagier formula and arithmetic applications. Ann. Math. 146(1), 111–147 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  2. Cornut, C., Vatsal, V.: CM points and quaternion algebras. Doc. Math. 10, 263–309 (2005) (electronic)

  3. Cornut, C., Vatsal, V.: Nontriviality of Rankin–Selberg L-functions and CM points. In: Burns, D., Buzzard, K., Nekovar, J. (eds). L-Functions and Galois Represenations, pp. 121–186. Cambridge University Press, Cambridge, MA (2007)

  4. Feigon B., Whitehouse D.: Averages of central L-values of Hilbert modular forms with an application to subconvexity. Duke Math. J. 149, 347–410 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. File, D., Martin, K., Pitale, A.: Test Vectors and Central L-Values for GL(2), arXiv:1310.1765 (2013)

  6. Gross B.: Local orders, root numbers and modular curves. Am. J. Math. 110, 1153–1182 (1988)

    Article  MATH  Google Scholar 

  7. Gross B., Prasad D.: Test vectors for linear forms. Math. Ann. 291, 243–355 (1991)

    MathSciNet  Article  Google Scholar 

  8. Hida H.: On p-adic L-functions of GL(2)\({\times}\) GL(2) over totally real fields. Ann. Inst. Fourier (Grenoble) 41, 311–391 (1991)

    MathSciNet  Article  Google Scholar 

  9. Jacquet H., Chen N.: Positivity of quadratic base change L-functions. Bulletin de la Societe Mathematique de France 129, 33–90 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Martin K., Whitehouse D.: Central L-values and toric periods for GL(2). IMRN 2009(1), 141–191 (2008)

    MathSciNet  Google Scholar 

  11. Pollack R., Weston T.: On anticyclotomic \({\mu}\)-invariants of modular forms. Compos. Math. 147, 1353–1381 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  12. Ratner M.: Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups. Duke Math. J. 77, 275–382 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  13. Vatsal V.: Uniform distribution of Heegner points. Invent. Math. 148, 1–46 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  14. Vatsal V.: Special values of anticyclotomic L-functions. Duke Math. J. 116(2), 219–261 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  15. Vatsal V.: Special value formulae for Rankin L-functions. MSRI Publ. 49, 165–190 (2004)

    MathSciNet  Google Scholar 

  16. Vignéras, M.-F.: Arithmétique des algèbres de quaternions. vol. 800, Springer Lecture Notes, (1980)

  17. Waldspurger J.: Sur les valeurs de certaines fonctions L automorphes en leur centre de symmétrie. Compos. Math. 54, 174–242 (1985)

    Google Scholar 

  18. Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83. Springer, Berlin (1996)

  19. Zhang S.: Gross–Zagier formula for GL(2). Asian J. Math. 5(2), 183–290 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Zhang S.: Heights of Heegner points on Shimura curves. Ann. Math. 153, 27–147 (2001)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Alia Hamieh.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hamieh, A. Special values of anticyclotomic L-functions modulo λ . manuscripta math. 145, 449–472 (2014).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification

  • Primary 11G40
  • Secondary 11G18
  • 11F67
  • 11F70