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Mathematics > Number Theory

arXiv:1803.10273 (math)
[Submitted on 27 Mar 2018 (v1), last revised 1 Dec 2023 (this version, v2)]

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

Authors:Zheng Liu, Giovanni Rosso
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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.
Subjects: Number Theory (math.NT)
Cite as: arXiv:1803.10273 [math.NT]
  (or arXiv:1803.10273v2 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1803.10273
Journal reference: Mathematische Annalen volume 378, pages153--231(2020)
Related DOI: https://doi.org/10.1007/s00208-020-01966-x

Submission history

From: Giovanni Rosso [view email]
[v1] Tue, 27 Mar 2018 18:59:39 UTC (62 KB)
[v2] Fri, 1 Dec 2023 01:01:47 UTC (91 KB)
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