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Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of p-adic L-functions

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Abstract

We study the derivative of the standard p-adic L-function associated with a P-ordinary Siegel modular form (for P a parabolic subgroup of \({{\,\mathrm{GL}\,}}(n)\)) when it presents a semi-stable trivial zero. This implies part of Greenberg’s conjecture on the order and leading coefficient of p-adic L-functions at such trivial zero. We use the method of Greenberg–Stevens. For the construction of the improved p-adic L-function we develop Hida theory for non-cuspidal Siegel modular forms.

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Acknowledgements

The authors thank Kai-Wen Lan, Vincent Pilloni, Eric Urban for many useful discussions, and the referees for their careful reading of the paper. Part of this work has been done while GR was a Herchel Smith fellow at Cambridge University and supernumerary fellow at Pembroke College, and during many visits at Columbia University; he would like to warmly thank these institutions. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1638352.

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  1. Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, USA

    Zheng Liu

  2. Departments of Mathematics and Statistics, Concordia University, Montreal, QC, Canada

    Giovanni Rosso

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  1. Zheng Liu
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Correspondence to Giovanni Rosso.

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Communicated by Wei Zhang.

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Liu, Z., Rosso, G. Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of p-adic L-functions. Math. Ann. 378, 153–231 (2020). https://doi.org/10.1007/s00208-020-01966-x

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