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An explicit Waldspurger formula for Hilbert modular forms II

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Abstract

We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. This formula extends our previous work, allowing to compute these central values when the main central value vanishes.

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References

  1. Böcherer, S., Schulze-Pillot, R.: On a theorem of Waldspurger and on Eisenstein series of Klingen type. Math. Ann. 288(3), 361–388 (1990). https://doi.org/10.1007/BF01444538

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997). https://doi.org/10.1006/jsco.1996.0125. Computational algebra and number theory (London, 1993)

  3. Gross, B.H.: Heights and the special values of \(L\)-series. In: Number theory (Montreal, Que., 1985). In: CMS Conference on Proceedings, vol. 7, pp. 115–187. American Mathematical Society, Providence, RI (1987)

  4. Gross, B.H.: Local orders, root numbers, and modular curves. Am. J. Math. 110(6), 1153–1182 (1988). https://doi.org/10.2307/2374689

    Article  MathSciNet  MATH  Google Scholar 

  5. Hiraga, K., Ikeda, T.: On the Kohnen plus space for Hilbert modular forms of half-integral weight I. Compos. Math. 149(12), 1963–2010 (2013). https://doi.org/10.1112/S0010437X13007276

    Article  MathSciNet  MATH  Google Scholar 

  6. Iwaniec, H.: Topics in classical automorphic forms. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1997). https://doi.org/10.1090/gsm/017

    Chapter  Google Scholar 

  7. Mao, Z.: On a generalization of Gross’s formula. Math. Z. 271(1–2), 593–609 (2012). https://doi.org/10.1007/s00209-011-0879-6

    Article  MathSciNet  MATH  Google Scholar 

  8. Mao, Z., Rodriguez-Villegas, F., Tornaría, G.: Computation of central value of quadratic twists of modular \(L\)-functions. Ranks of Elliptic Curves and Random Matrix Theory. London Mathematical Society, Lecture Note Series, vol. 341, pp. 273–288. Cambridge University Press, Cambridge (2007)

    Chapter  Google Scholar 

  9. Roberts, D.: Shimura curves analogous to X(0)(N) (1989). Ph.D. Thesis, Harvard University

  10. Shimura, G.: On Hilbert modular forms of half-integral weight. Duke Math. J. 55(4), 765–838 (1987). https://doi.org/10.1215/S0012-7094-87-05538-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Sirolli, N.: Preimages for the Shimura map on Hilbert modular forms. J. Number Theory 145, 79–98 (2014). https://doi.org/10.1016/j.jnt.2014.05.006

    Article  MathSciNet  MATH  Google Scholar 

  12. Sirolli, N., Tornaría, G.: An explicit Waldspurger formula for Hilbert modular forms. Trans. Am. Math. Soc. 370(9), 6153–6168 (2018). https://doi.org/10.1090/tran/7112

    Article  MathSciNet  MATH  Google Scholar 

  13. Sirolli, N., Tornaría, G.: Central values of twisted \({L}\)-series attached to Hilbert modular forms, using weight functions. (2020). http://www.cmat.edu.uy/cnt/hmf

  14. Stein, W., et al.: Sage Mathematics Software (Version 9.0). The Sage Development Team (2020). http://www.sagemath.org

  15. Su, R.H.: Newforms in the Kohnen plus space. J. Number Theory 184, 133–178 (2018). https://doi.org/10.1016/j.jnt.2017.08.015

    Article  MathSciNet  MATH  Google Scholar 

  16. Tornaría, G.: The Brandt module of ternary quadratic lattices (2005). Ph.D. Thesis, The University of Texas at Austin

  17. Waldspurger, J.L.: Correspondances de Shimura et quaternions. Forum Math. 3(3), 219–307 (1991). https://doi.org/10.1515/form.1991.3.219

    Article  MathSciNet  MATH  Google Scholar 

  18. Xue, H.: Central values of Rankin \(L\)-functions. Int. Math. Res. Not. (2006). https://doi.org/10.1155/IMRN/2006/26150

    Article  MathSciNet  MATH  Google Scholar 

  19. Xue, H.: Central values of \(L\)-functions and half-integral weight forms. Proc. Am. Math. Soc. 139(1), 21–30 (2011). https://doi.org/10.1090/S0002-9939-2010-10660-3

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, S.W.: Gross-Zagier formula for \({\rm GL}_2\). Asian J. Math. 5(2), 183–290 (2001)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Universidad de Buenos Aires and IMAS-CONICET, Buenos Aires, Argentina

    Nicolás Sirolli

  2. Universidad de la República, Montevideo, Uruguay

    Gonzalo Tornaría

Authors
  1. Nicolás Sirolli
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  2. Gonzalo Tornaría
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Correspondence to Nicolás Sirolli.

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Sirolli, N., Tornaría, G. An explicit Waldspurger formula for Hilbert modular forms II. Ramanujan J 55, 1189–1212 (2021). https://doi.org/10.1007/s11139-020-00280-z

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