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Anticyclotomic p-adic L-functions and Ichino’s formula

  • Dan J. CollinsEmail author
Article
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Abstract

We give a new construction of a p-adic L-function \({\mathcal {L}}(f,\Xi )\), for f a holomorphic newform and \(\Xi \) an anticyclotomic family of Hecke characters of \(\mathbb {Q}(\sqrt{-d})\). The construction uses Ichino’s triple product formula to express the central values of \(L(f,\xi ,s)\) in terms of Petersson inner products, and then uses results of Hida to interpolate them. The resulting construction is well-suited for studying what happens when f is replaced by a modular form congruent to it modulo p, and has future applications in the case where f is residually reducible.

Keywords

p-Adic L-functions Triple-product L-functions Hida families 

Résumé

Nous donnons une nouvelle construction d’une fonction Lp-adique \({\mathcal {L}}(f,\Xi )\), pour f une forme primitive holomorphe et \(\Xi \) une famille anticyclotomique de caractères de Hecke de \(\mathbb {Q}(\sqrt{-d})\). La construction utilise la formule du produit triple d’Ichino pour exprimer les valeurs centrales de \(L(f,\xi ,s)\) en terme de produits scalaires de Petersson, et certains résultats de Hida pour les interpoler p-adiquement. La construction qui en découle permet d’étudier ce qui se passe quand f est remplacée par une forme modulaire qui lui est congruente modulo p, et a des applications futures dans le cas où f est résiduellement réductible.

Mathematics Subject Classification

Primary 11F67 

Notes

Acknowledgements

I would like to thank my advisor Christopher Skinner for suggesting this project to me and for offering insights and encouragement, and to thank Peter Humphries and Vinayak Vatsal for helpful conversations. Much of this research was completed while the author was supported by an NSF Graduate Research Fellowship (DGE-1148900).

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Copyright information

© Fondation Carl-Herz and Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MilanMilanItaly

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