Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Kohnen's formula and a conjecture of Darmon and Tornaría


Authors: Matteo Longo and Zhengyu Mao
Journal: Trans. Amer. Math. Soc. 370 (2018), 73-98
MSC (2010): Primary 11F37, 11F67, 11G40; Secondary 11F85, 14G05
DOI: https://doi.org/10.1090/tran/6930
Published electronically: May 16, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We generalize a result of W. Kohnen (1985) to explicit Waldspurger lifts constructed by E. M. Baruch and Z. Mao (2007). As an application, we prove a conjecture formulated by H. Darmon and G. Tornaría (2008).


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

  • [1] Ehud Moshe Baruch and Zhengyu Mao, Central value of automorphic $ L$-functions, Geom. Funct. Anal. 17 (2007), no. 2, 333-384. MR 2322488, https://doi.org/10.1007/s00039-007-0601-3
  • [2] Massimo Bertolini and Henri Darmon, Hida families and rational points on elliptic curves, Invent. Math. 168 (2007), no. 2, 371-431. MR 2289868, https://doi.org/10.1007/s00222-007-0035-4
  • [3] Massimo Bertolini and Henri Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields, Ann. of Math. (2) 170 (2009), no. 1, 343-370. MR 2521118, https://doi.org/10.4007/annals.2009.170.343
  • [4] Massimo Bertolini, Henri Darmon, and Samit Dasgupta, Stark-Heegner points and special values of $ L$-series, $ L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 1-23. MR 2392351, https://doi.org/10.1017/CBO9780511721267.002
  • [5] Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Nonvanishing theorems for $ L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543-618. MR 1074487, https://doi.org/10.1007/BF01233440
  • [6] Henri Darmon, Integration on $ \mathcal {H}_p\times \mathcal {H}$ and arithmetic applications, Ann. of Math. (2) 154 (2001), no. 3, 589-639. MR 1884617, https://doi.org/10.2307/3062142
  • [7] Henri Darmon and Gonzalo Tornaría, Stark-Heegner points and the Shimura correspondence, Compos. Math. 144 (2008), no. 5, 1155-1175. MR 2457522, https://doi.org/10.1112/S0010437X08003552
  • [8] Roger Godement, Notes on jacquet langlands' theory, Institute for Advanced Study, 1970.
  • [9] Matthew Greenberg, Marco Adamo Seveso, and Shahab Shahabi, Modular $ p$-adic $ L$-functions attached to real quadratic fields and arithmetic applications, J. Reine Angew. Math. 721 (2016), 167-231. MR 3574881, https://doi.org/10.1515/crelle-2014-0088
  • [10] Matthew Greenberg, Stark-Heegner points and the cohomology of quaternionic Shimura varieties, Duke Math. J. 147 (2009), no. 3, 541-575. MR 2510743, https://doi.org/10.1215/00127094-2009-017
  • [11] Ralph Greenberg and Glenn Stevens, $ p$-adic $ L$-functions and $ p$-adic periods of modular forms, Invent. Math. 111 (1993), no. 2, 407-447. MR 1198816, https://doi.org/10.1007/BF01231294
  • [12] Guhanvenkat Harikumar, Darmon cycles and kohnen-shintani lifting, 2015, Thesis (Ph.D.)-University of Bordeaux and University of Padova.
  • [13] Winfried Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237-268. MR 783554, https://doi.org/10.1007/BF01455989
  • [14] Matteo Longo and Marc-Hubert Nicole, The $ \Lambda $-adic Shimura-Shintani-Waldspurger correspondence, Doc. Math. 18 (2013), 1-21. MR 3035767
  • [15] Matteo Longo and Marc-Hubert Nicole, The Saito-Kurokawa lifting and Darmon points, Math. Ann. 356 (2013), no. 2, 469-486. MR 3048604, https://doi.org/10.1007/s00208-012-0846-5
  • [16] Matteo Longo, Victor Rotger, and Stefano Vigni, On rigid analytic uniformizations of Jacobians of Shimura curves, Amer. J. Math. 134 (2012), no. 5, 1197-1246. MR 2975234, https://doi.org/10.1353/ajm.2012.0033
  • [17] Matteo Longo, Victor Rotger, and Stefano Vigni, Special values of $ L$-functions and the arithmetic of Darmon points, J. Reine Angew. Math. 684 (2013), 199-244. MR 3181561
  • [18] Matteo Longo and Stefano Vigni, The rationality of quaternionic Darmon points over genus fields of real quadratic fields, Int. Math. Res. Not. IMRN 13 (2014), 3632-3691. MR 3229764
  • [19] Zhengyu Mao, On a generalization of Gross's formula, Math. Z. 271 (2012), no. 1-2, 593-609. MR 2917160, https://doi.org/10.1007/s00209-011-0879-6
  • [20] Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modulo $ l$, Ann. of Math. (2) 147 (1998), no. 2, 453-470. MR 1626761, https://doi.org/10.2307/121015
  • [21] Ken Ono and Christopher Skinner, Non-vanishing of quadratic twists of modular $ L$-functions, Invent. Math. 134 (1998), no. 3, 651-660. MR 1660945, https://doi.org/10.1007/s002220050275
  • [22] Jeehoon Park, $ p$-adic family of half-integral weight modular forms via overconvergent Shintani lifting, Manuscripta Math. 131 (2010), no. 3-4, 355-384. MR 2592085, https://doi.org/10.1007/s00229-009-0323-y
  • [23] Alexandru A. Popa, Central values of Rankin $ L$-series over real quadratic fields, Compos. Math. 142 (2006), no. 4, 811-866. MR 2249532, https://doi.org/10.1112/S0010437X06002259
  • [24] Kartik Prasanna, Arithmetic properties of the Shimura-Shintani-Waldspurger correspondence, Invent. Math. 176 (2009), no. 3, 521-600. With an appendix by Brian Conrad. MR 2501296, https://doi.org/10.1007/s00222-008-0169-z
  • [25] Kartik Prasanna, On the Fourier coefficients of modular forms of half-integral weight, Forum Math. 22 (2010), no. 1, 153-177. MR 2604368, https://doi.org/10.1515/FORUM.2010.008
  • [26] Victor Rotger and Marco Adamo Seveso, $ \mathcal {L}$-invariants and Darmon cycles attached to modular forms, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 6, 1955-1999. MR 2984593, https://doi.org/10.4171/JEMS/352
  • [27] Marco Adamo Seveso, Congruences and rationality of Stark-Heegner points, J. Number Theory 132 (2012), no. 3, 414-447. MR 2875348, https://doi.org/10.1016/j.jnt.2011.10.001
  • [28] Marco Adamo Seveso, $ p$-adic $ L$-functions and the rationality of Darmon cycles, Canad. J. Math. 64 (2012), no. 5, 1122-1181. MR 2979580, https://doi.org/10.4153/CJM-2011-076-8
  • [29] Marco Adamo Seveso, Heegner cycles and derivatives of $ p$-adic $ L$-functions, J. Reine Angew. Math. 686 (2014), 111-148. MR 3176601, https://doi.org/10.1515/crelle-2012-0027
  • [30] Shahab Shahabi, p-adic deformation of Shintani cycles, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)-McGill University (Canada). MR 2713606
  • [31] Takuro Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83-126. MR 0389772
  • [32] Glenn Stevens, $ \Lambda $-adic modular forms of half-integral weight and a $ \Lambda $-adic Shintani lifting, Arithmetic geometry (Tempe, AZ, 1993) Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 129-151. MR 1299739, https://doi.org/10.1090/conm/174/01856
  • [33] J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1-132 (French). MR 577010
  • [34] 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
  • [35] J.-L. Waldspurger, Sur les valeurs de certaines fonctions $ L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173-242 (French). MR 783511
  • [36] Jean-Loup Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219-307 (French). MR 1103429, https://doi.org/10.1515/form.1991.3.219

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F37, 11F67, 11G40, 11F85, 14G05

Retrieve articles in all journals with MSC (2010): 11F37, 11F67, 11G40, 11F85, 14G05


Additional Information

Matteo Longo
Affiliation: Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: mlongo@math.unipd.it

Zhengyu Mao
Affiliation: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102
Email: zmao@rutgers.edu

DOI: https://doi.org/10.1090/tran/6930
Received by editor(s): November 25, 2014
Received by editor(s) in revised form: February 14, 2016, and February 29, 2016
Published electronically: May 16, 2017
Additional Notes: The first author was partly supported by PRIN 2010-11, Cariparo Foundation Project Differential Methods in Arithmetic, Geometry and Algebra, PRAT 2013 and INDAM. The second author was partly supported by NSF DMS 1400063.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society