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Berger, T. orcid.org/0000-0002-5207-6221 and Betina, A. (Submitted: 2019) On Siegel eigenvarieties at Saito-Kurokawa points. arXiv. (Submitted)
Abstract
We study the geometry of the Siegel eigenvariety EΔ of paramodular tame level Δ associated to a squarefree N∈N+ at certain points having a critical slope. For k≥2 let f be a cuspidal eigenform of S2k−2(Γ0(N)) ordinary at a prime p∤N with sign ϵf=−1 and write α for the unit root of the Hecke polynomial of f at p. Let SK(f)α be the semi-ordinary p-stabilization of the Saito-Kurokawa lift of the cusp form f to GSp(4) of weight (k,k) of tame level Δ. Under the assumption that the dimension of the Selmer group H1f,unr(Q,ρf(k−1)) attached to f is at most one and some mild assumptions on the mod p representation ρ¯f associated to f, we show that the rigid analytic space EΔ is smooth at the point x corresponding to SK(f)α. This means that there exists a unique irreducible component of EΔ specializing to x, and we also show that this irreducible component is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions on ρ¯f that the smoothness failure of EΔ at x yields that dimH1f,unr(Q,ρf(k−1))≥2.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019 The Author(s). For reuse permissions, please contact the Author(s). |
Keywords: | math.NT; math.NT; 11F33, 11F46, 11F80, 14G22 |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 15 Oct 2019 08:18 |
Last Modified: | 11 Nov 2022 16:57 |
Status: | Submitted |
Identification Number: | 10.48550/arXiv.1902.05885 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:152069 |
Available Versions of this Item
- On Siegel eigenvarieties at Saito-Kurokawa points. (deposited 15 Oct 2019 08:18) [Currently Displayed]