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Inventiones mathematicae

, Volume 196, Issue 1, pp 69–138 | Cite as

On special zeros of p-adic L-functions of Hilbert modular forms

  • Michael SpießEmail author
Article

Abstract

Let E be a modular elliptic curve over a totally real number field F. We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain p-adic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the \(\mathcal {L}\)-invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.

Mathematics Subject Classification (2000)

11F41 11F67 11F70 11G40 

Notes

Acknowledgements

I thank Vytautas Paskunas for several helpful conversations and Kumar Murty for providing me with the reference [12]. I am grateful to H. Deppe, L. Gehrmann, S. Molina and M. Seveso for useful comments on an earlier draft. Also the referee suggested several useful improvements.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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