Annales mathématiques du Québec

, Volume 40, Issue 2, pp 303–324 | Cite as

Big Heegner points and special values of L-series

  • Francesc Castella
  • Matteo Longo


In Longo and Vigni (Manuscr Math 135:273–328, 2011), Howard’s construction of big Heegner points on modular curves was extended to general Shimura curves over the rationals. In this paper, we relate the higher weight specializations of the big Heegner points of Longo and Vigni (Manuscr Math 135:273–328, 2011) in the definite setting to certain higher weight analogues of the Bertolini–Darmon theta elements (Bertolini and Darmon in Invent Math 126:413–456, 1996). As a consequence of this relation, some of the conjectures in Longo and Vigni (Manuscr Math 135:273–328, 2011) are deduced from recent results of Chida and Hsieh (J Reine Angew Math, 2015).


Modular forms Hida families P-adic L-functions 


L’article Longo et Vigni (Manuscr Math 135: 273-328, 2011) généralise la construction de Howard de “Big Heegner points” sur les courbes modulaires aux courbes de Shimura sur les nombres rationnels. Dans cet article, nous faisons le lien entre les spécialisations de poids plus grand que 2 des “Big Heegner points” de Longo et Vigni (Manuscr Math 135:273–328, 2011) dans le cadre défini et certains analogues en poids plus grand que 2 des éléments thêta de Bertolini-Darmon (Bertolini et Darmon dans Invent Math 126:413–456, 1996). En conséquence de cette relation, certaines des conjectures dans Longo et Vigni (Manuscr Math 135:273–328, 2011) sont déduites des résultats récents de Chida et Hsieh (J Reine Angew Math, 2015).

Mathematics Subject Classification

11F11 11F33 11618 



The authors would like to thank Ming-Lun Hsieh for several helpful conversations related to this work, and the anonymous referee for a number of useful comments and suggestions.


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© Fondation Carl-Herz and Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Dipartimento di MatematicaUniversità di PadovaPadovaItaly

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