Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Central values of $ L$-functions and half-integral weight forms


Author: Hui Xue
Journal: Proc. Amer. Math. Soc. 139 (2011), 21-30
MSC (2010): Primary 11F67, 11F37
DOI: https://doi.org/10.1090/S0002-9939-2010-10660-3
Published electronically: August 26, 2010
MathSciNet review: 2729067
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a relation between the Fourier coefficients of certain Hilbert modular forms of half-integral weight and central values of the corresponding Rankin $ L$-functions. The result generalizes the classical theorem by Waldspurger. The approach is geometric and generalizes that of Gross and Hatcher.


References [Enhancements On Off] (What's this?)

  • 1. J. Brzezinski, On embedding numbers into quaternion orders, Comment. Math. Helv. 66 (1991), no. 2, 302-318. MR 1107843 (92e:11133)
  • 2. B. Gross, Heights and the special values of $ L$-series, Number theory (Montreal, Que., 1985), CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115-187. MR 894322 (89c:11082)
  • 3. -, Local orders, root numbers, and modular curves, Amer. J. Math. 110 (1988), no. 6, 1153-1182. MR 970123 (90b:11053)
  • 4. B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of $ L$-series. II, Math. Ann. 278 (1987), no. 1-4, 497-562. MR 909238 (89i:11069)
  • 5. R. Hatcher, Heights and $ L$-series, Canad. J. Math. 42 (1990), no. 3, 533-560. MR 1062744 (92b:11031)
  • 6. -, $ L$-series and modular forms of half-integral weight, Proc. Amer. Math. Soc. 122 (1994), no. 3, 683-688. MR 1233973 (95a:11045)
  • 7. H. Hijikata, A. Pizer, and T. Shemanske, Orders in quaternion algebras, J. Reine Angew. Math. 394 (1989), 59-106. MR 977435 (90d:11128)
  • 8. W. Pfetzer, Die Wirkung der Modulsubstitutionen auf mehrafache Thetareihen zu quadratischen Formen ungerader Variablenzahl, Arch. Math. (Basel) 4 (1953), 448-454. MR 0059945 (15:603d)
  • 9. A. Pizer, An algorithm for computing modular forms on $ \Gamma _{0}(N)$, J. Algebra 64 (1980), no. 2, 340-390. MR 579066 (83g:10020)
  • 10. G. Shimura, On the Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math. J. 71 (1993), no. 2, 501-557. MR 1233447 (94e:11046)
  • 11. -, On the transformation formulas of theta series, Amer. J. Math. 115 (1993), no. 5, 1011-1052. MR 1246183 (94h:11045)
  • 12. J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375-484. MR 646366 (83h:10061)
  • 13. H. Xue, Central values of Rankin $ L$-functions, Int. Math. Res. Not. (2006), Art. ID 26150, 41. MR 2249999 (2008e:11059)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F67, 11F37

Retrieve articles in all journals with MSC (2010): 11F67, 11F37


Additional Information

Hui Xue
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634
Email: huixue@clemson.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10660-3
Received by editor(s): January 12, 2010
Published electronically: August 26, 2010
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society