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Transactions of the American Mathematical Society

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An explicit Waldspurger formula for Hilbert modular forms

Authors: Nicolás Sirolli and Gonzalo Tornaría
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F67, 11F41, 11F37
Published electronically: March 20, 2018
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We describe a construction of preimages for the Shimura map on Hilbert modular forms and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted $ L$-functions. Our construction is inspired by that of Gross and applies to any nontrivial level and arbitrary base field, subject to certain conditions on the Atkin-Lehner eigenvalues and on the weight.

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  • [BM07] Ehud Moshe Baruch and Zhengyu Mao, Central value of automorphic $ L$-functions, Geom. Funct. Anal. 17 (2007), no. 2, 333-384. MR 2322488,
  • [BSP90] Siegfried Böcherer and Rainer Schulze-Pillot, On a theorem of Waldspurger and on Eisenstein series of Klingen type, Math. Ann. 288 (1990), no. 3, 361-388. MR 1079868,
  • [CST14] Li Cai, Jie Shu, and Ye Tian, Explicit Gross-Zagier and Waldspurger formulae, Algebra Number Theory 8 (2014), no. 10, 2523-2572. MR 3298547,
  • [Geb09] Ute Gebhardt,
    Explicit construction of spaces of Hilbert modular cusp forms using quaternionic theta series,
    Thesis (Ph.D.)-Universitat des Saarlandes, 2009.
  • [Gro87] Benedict H. 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
  • [Gro88] Benedict H. Gross, Local orders, root numbers, and modular curves, Amer. J. Math. 110 (1988), no. 6, 1153-1182. MR 970123,
  • [HI13] Kaoru Hiraga and Tamotsu Ikeda, On the Kohnen plus space for Hilbert modular forms of half-integral weight I, Compos. Math. 149 (2013), no. 12, 1963-2010. MR 3143703,
  • [Hij74] Hiroaki Hijikata, Explicit formula of the traces of Hecke operators for $ \Gamma_{0}(N)$, J. Math. Soc. Japan 26 (1974), 56-82. MR 0337783,
  • [Iwa97] Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964
  • [Koh82] Winfried Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32-72. MR 660784,
  • [Mao12] Zhengyu Mao, On a generalization of Gross's formula, Math. Z. 271 (2012), no. 1-2, 593-609. MR 2917160,
  • [MRVT07] Z. Mao, F. Rodriguez-Villegas, and G. Tornaría, Computation of central value of quadratic twists of modular $ L$-functions, Ranks of elliptic curves and random matrix theory, London Math. Soc. Lecture Note Ser., vol. 341, Cambridge Univ. Press, Cambridge, 2007, pp. 273-288. MR 2322352,
  • [Piz76] Arnold Pizer, On the arithmetic of quaternion algebras. II, J. Math. Soc. Japan 28 (1976), no. 4, 676-688. MR 0432600,
  • [Rob89] David Peter Roberts, Shimura curves analogous to $ X_0(N)$, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-Harvard University, 1989. MR 2637583
  • [RTV14] Nathan C. Ryan, Gonzalo Tornaría, and John Voight, Nonvanishing of twists of $ L$-functions attached to Hilbert modular forms, LMS J. Comput. Math. 17 (2014), no. suppl. A, 330-348. MR 3240813,
  • [Shi87] Goro Shimura, On Hilbert modular forms of half-integral weight, Duke Math. J. 55 (1987), no. 4, 765-838. MR 916119,
  • [Shi93] Goro Shimura, On the Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math. J. 71 (1993), no. 2, 501-557. MR 1233447,
  • [Sir14] Nicolás Sirolli, Preimages for the Shimura map on Hilbert modular forms, J. Number Theory 145 (2014), 79-98. MR 3253294,
  • [Vig80] Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
  • [Wal81] 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 (French). MR 646366
  • [Wal91] Jean-Loup Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219-307 (French). MR 1103429,
  • [Xue06] Hui Xue, Central values of Rankin $ L$-functions, Int. Math. Res. Not. , posted on (2006), Art. ID 26150, 41 pp.. MR 2249999,
  • [Xue11] Hui Xue, Central values of $ L$-functions and half-integral weight forms, Proc. Amer. Math. Soc. 139 (2011), no. 1, 21-30. MR 2729067,
  • [Zha01] Shou-Wu Zhang, Gross-Zagier formula for $ {\rm GL}_2$, Asian J. Math. 5 (2001), no. 2, 183-290. MR 1868935,

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Additional Information

Nicolás Sirolli
Affiliation: Departamento de Matemática, Oficina 2096, Facultad de Ciencias Exactas y Naturales (C1428EGA) Pabellón I, Ciudad Universitaria, Ciudad Autónoma de Buenos Aires, Argentina

Gonzalo Tornaría
Affiliation: Centro de Matematica, Universidad de la República, 11400 Montevideo, Uruguay

Keywords: Waldspurger formula, Hilbert modular forms, Shimura correspondence
Received by editor(s): March 30, 2016
Received by editor(s) in revised form: October 26, 2016
Published electronically: March 20, 2018
Additional Notes: The first author was fully supported by the ANII under grant code PD_NAC_2013_1_11010
Article copyright: © Copyright 2018 by Nicolás Sirolli and Gonzalo Tornaría

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