On Siegel eigenvarieties at Saito–Kurokawa points
Annales de l'Institut Fourier, Online first, 61 p.

We study the geometry of the p-adic Siegel eigenvariety of paramodular tame level at certain Saito–Kurokawa points having a critical slope. For k2 let f be a cuspidal new eigenform of S 2k-2 (Γ 0 (N)) ordinary at a prime pN 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 H f,unr 1 (,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 is smooth at the point corresponding to π α , and that the irreducible component of 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 Δ at π α yields that dimH f,unr 1 (,ρ f (k-1))2.

On étudie la géométrie de la variété de Hecke-Siegel Δ de niveau paramodulaire Δ associée à un entier N en certains points de Saito-Kurokawa dont la pente est critique. Pour k2, soit f une forme cuspidale de S 2k-2 (Γ 0 (N)), ordinaire en un nombre premier pN, de signe ϵ f =-1, et notons α l’unité p-adique racine du polynôme de Hecke de f en p. Soit π α la p-stabilisation semi-ordinaire du relèvement de Saito–Kurokawa de f à GSp 4 de poids (k,k) et de niveau modéré Δ. Sous l’hypothèse que la dimension du groupe de Selmer H f,unr 1 (,ρ f (k-1)) associé à f est au plus égale à 1 et certaines hypothèses supplémentaires sur la représentation p-adique ρ f associée à f, on démontre que l’espace analytique rigide Δ est lisse au point correspondant à π α . Cela signifie qu’il existe une unique composante irréductible de Δ passant par π α et on démontre que cette composante n’est pas globalement endoscopique. Pour finir, on donne une application à la conjecture de Bloch-Kato, en prouvant, sous certaines hypothèses raisonnables, qu’une singularité de Δ en π α entraîne que dimH f,unr 1 (,ρ f (k-1))2.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3482
Classification: 11F33,  11F46,  11F80,  14G22
Keywords: Eigenvarieties for GSp 4 , Saito–Kurokawa lift, Selmer groups and pseudo-deformation theory
Berger, Tobias 1; Betina, Adel 2

1 School of Mathematics and Statistics University of Sheffield Hicks Building, Hounsfield Road Sheffield S3 7RH (UK)
2 Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 A-1090 Wien (Austria)
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Berger, Tobias; Betina, Adel. On Siegel eigenvarieties at Saito–Kurokawa points. Annales de l'Institut Fourier, Online first, 61 p.

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