On Siegel eigenvarieties at Saito–Kurokawa points

Berger, Tobias; Betina, Adel

doi:10.5802/aif.3482

On Siegel eigenvarieties at Saito–Kurokawa points
[Les Variétés de Hecke pour

{GSp}_{4}

aux points de Saito–Kurokawa]

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)

Annales de l'Institut Fourier, Tome 72 (2022) no. 3, pp. 901-961.

Résumé
Abstract

On étudie la géométrie de la variété de Hecke-Siegel $ℰ_{Δ}$ de niveau paramodulaire $Δ$ associée à un entier $N$ en certains points de Saito-Kurokawa dont la pente est critique. Pour $k \geq 2$ , soit $f$ une forme cuspidale de $S_{2 k - 2} (Γ_{0} (N))$ , ordinaire en un nombre premier $p ∤ N$ , 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 $H_{f, unr}^{1} (ℚ, ρ_{f} (k - 1))$ 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 $ℰ_{Δ}$ est lisse au point correspondant à $π_{α}$ . Cela signifie qu’il existe une unique composante irréductible de $ℰ_{Δ}$ 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 $ℰ_{Δ}$ en $π_{α}$ entraîne que $dim H_{f, unr}^{1} (ℚ, ρ_{f} (k - 1)) \geq 2$ .

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 $k \geq 2$ let $f$ be a cuspidal new eigenform of $S_{2 k - 2} (Γ_{0} (N))$ ordinary at a prime $p ∤ N$ 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 $H_{f, unr}^{1} (ℚ, ρ_{f} (k - 1))$ 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 $ℰ_{Δ}$ at $π_{α}$ yields that $dim H_{f, unr}^{1} (ℚ, ρ_{f} (k - 1)) \geq 2$ .

Reçu le : 2019-08-15
Révisé le : 2020-02-26
Accepté le : 2020-05-13
Publié le : 2022-09-12

DOI : 10.5802/aif.3482

Classification : 11F33, 11F46, 11F80, 14G22
Keywords: Eigenvarieties for $\mathrm{GSp}_4$, Saito–Kurokawa lift, Selmer groups and pseudo-deformation theory
Mot clés : Variétés de Hecke pour $\mathrm{GSp}_4$, formes de Saito–Kurokawa, groupes de Selmer et Théorie des déformations

Affiliations des auteurs :

Berger, Tobias ¹ ; Betina, Adel ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Berger, Tobias and Betina, Adel},
     title = {On {Siegel} eigenvarieties at {Saito{\textendash}Kurokawa} points},
     journal = {Annales de l'Institut Fourier},
     pages = {901--961},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
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     year = {2022},
     doi = {10.5802/aif.3482},
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Berger, Tobias; Betina, Adel. On Siegel eigenvarieties at Saito–Kurokawa points. Annales de l'Institut Fourier, Tome 72 (2022) no. 3, pp. 901-961. doi : 10.5802/aif.3482. https://aif.centre-mersenne.org/articles/10.5802/aif.3482/

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