On Siegel eigenvarieties at Saito-Kurokawa points

2019/02/15 by Berger, Tobias, Betina, Adel
#11F33 #11F46 #11F80 #14G22 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1902.05885

Abstract

We study the geometry of the p-adic Siegel eigenvariety E of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For k ≥ 2 let f be a cuspidal new eigenform of S_2k-2(Γ₀(N)) ordinary at a prime p\nmid N with sign ε_f=-1 and write α for the p-adic unit root of the Hecke polynomial of f at p. Let π_α be the semi-ordinary p-stabilization of the Saito-Kurokawa lift of the cusp form f to GSp(4) of weight (k,k) and paramodular tame level. Under the assumption that the dimension of the Selmer group H¹_f,unr(ℚ,ρ_f(k-1)) attached to f is at most one and some mild assumptions on the automorphic representation attached to f, we show that E is smooth at the point corresponding to π_α, and that the irreducible component of E specializing to π_α is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions that the smoothness failure of E at π_α yields that dim H¹_f,unr(ℚ,ρ_f(k-1))≥ 2.

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