We study the geometry of the -adic Siegel eigenvariety of paramodular tame level at certain Saito–Kurokawa points having a critical slope. For let be a cuspidal new eigenform of ordinary at a prime with sign and write for the -adic unit root of the Hecke polynomial of at . Let be the semi-ordinary -stabilization of the Saito–Kurokawa lift of the cusp form to of weight and paramodular tame level. Under the assumption that the dimension of the Selmer group attached to is at most one and some mild assumptions on the automorphic representation attached to , we show that is smooth at the point corresponding to , and that the irreducible component of specializing to 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 at yields that .
On étudie la géométrie de la variété de Hecke-Siegel de niveau paramodulaire associée à un entier en certains points de Saito-Kurokawa dont la pente est critique. Pour , soit une forme cuspidale de , ordinaire en un nombre premier , de signe , et notons l’unité -adique racine du polynôme de Hecke de en . Soit la -stabilisation semi-ordinaire du relèvement de Saito–Kurokawa de à de poids et de niveau modéré . Sous l’hypothèse que la dimension du groupe de Selmer associé à est au plus égale à et certaines hypothèses supplémentaires sur la représentation -adique associée à , on démontre que l’espace analytique rigide est lisse au point correspondant à . Cela signifie qu’il existe une unique composante irréductible de passant par 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 en entraîne que .
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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
Berger, Tobias 1; Betina, Adel 2
@article{AIF_2022__72_3_901_0, 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}, number = {3}, year = {2022}, doi = {10.5802/aif.3482}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3482/} }
TY - JOUR AU - Berger, Tobias AU - Betina, Adel TI - On Siegel eigenvarieties at Saito–Kurokawa points JO - Annales de l'Institut Fourier PY - 2022 SP - 901 EP - 961 VL - 72 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3482/ DO - 10.5802/aif.3482 LA - en ID - AIF_2022__72_3_901_0 ER -
%0 Journal Article %A Berger, Tobias %A Betina, Adel %T On Siegel eigenvarieties at Saito–Kurokawa points %J Annales de l'Institut Fourier %D 2022 %P 901-961 %V 72 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3482/ %R 10.5802/aif.3482 %G en %F AIF_2022__72_3_901_0
Berger, Tobias; Betina, Adel. On Siegel eigenvarieties at Saito–Kurokawa points. Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 901-961. doi : 10.5802/aif.3482. https://aif.centre-mersenne.org/articles/10.5802/aif.3482/
[1] -adic families of Siegel modular cuspforms, Ann. Math., Volume 181 (2015) no. 2, pp. 623-697 | DOI | MR | Zbl
[2] Automorphic representations of , Contributions to automorphic forms, geometry, and number theory, Johns Hopkins University Press, 2004, pp. 65-81 | MR | Zbl
[3] Nonsmooth classical points on eigenvarieties, Duke Math. J., Volume 145 (2008) no. 1, pp. 71-90 | DOI | MR | Zbl
[4] Eigenvarieties and -adic -functions (2010) (book in preparation, http://people.brandeis.edu/~jbellaic/preprint/coursebook.pdf)
[5] Critical -adic -functions, Invent. Math., Volume 189 (2012) no. 1, pp. 1-60 | DOI | MR | Zbl
[6] Lissité de la courbe de Hecke de aux points Eisenstein critiques, J. Inst. Math. Jussieu, Volume 5 (2006) no. 2, pp. 333-349 | DOI | MR | Zbl
[7] Families of Galois representations and Selmer groups, Astérisque, 324, Société Mathématique de France, 2009, xii+314 pages | MR
[8] On the eigencurve at classical weight 1 points, Duke Math. J., Volume 165 (2016) no. 2, pp. 245-266 | DOI | MR | Zbl
[9] Deformations of Saito–Kurokawa type and the paramodular conjecture, Am. J. Math., Volume 142 (2020) no. 6, pp. 1821-1876 | DOI | MR | Zbl
[10] Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci. (1993) no. 78, pp. 5-161 | DOI | Numdam | MR | Zbl
[11] Geometry of the eigencurve at CM points and trivial zeros of Katz -adic -functions https://arxiv.org/abs/1907.09422, to appear in Adv. Math. 384 (2021)
[12] On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve (https://arxiv.org/abs/1804.00648, to appear in Am. J. Math)
[13] Familles -adiques de formes automorphes pour , J. Reine Angew. Math., Volume 570 (2004), pp. 143-217 | DOI | MR | Zbl
[14] , , and overconvergence, Int. Math. Res. Not. (1995) no. 1, pp. 23-41 | DOI | MR | Zbl
[15] The eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996) (London Mathematical Society Lecture Note Series), Volume 254, Cambridge University Press, 1998, pp. 1-113 | DOI | MR | Zbl
[16] Irreducible components of rigid spaces, Ann. Inst. Fourier, Volume 49 (1999) no. 2, pp. 473-541 | DOI | Numdam | MR | Zbl
[17] Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 22, Springer, 1990, xii+316 pages (With an appendix by David Mumford) | DOI | MR
[18] Périodes -adiques, Astérisque, 223 (1994), pp. 1-397 (Papers from the seminar held in Bures-sur-Yvette, 1988) | MR
[19] Arthur’s multiplicity formula for and restriction to , J. Éc. Polytech., Math., Volume 6 (2019), pp. 469-535 | Numdam | MR | Zbl
[20] Families of Picard modular forms and an application to the Bloch–Kato conjecture, Compos. Math., Volume 155 (2019) no. 7, pp. 1327-1401 | DOI | MR | Zbl
[21] Galois representations into attached to ordinary cusp forms, Invent. Math., Volume 85 (1986) no. 3, pp. 545-613 | DOI | MR | Zbl
[22] Control theorems of coherent sheaves on Shimura varieties of PEL type, J. Inst. Math. Jussieu, Volume 1 (2002) no. 1, pp. 1-76 | DOI | MR | Zbl
[23] -adic Hodge theory and values of zeta functions of modular forms, Cohomologies -adiques et applications arithmétiques. III (Astérisque), Volume 295, Société Mathématique de France, 2004, pp. ix, 117-290 | Numdam | MR | Zbl
[24] Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math., Volume 153 (2003) no. 2, pp. 373-454 | DOI | MR | Zbl
[25] Geometric deformations of modular Galois representations, Invent. Math., Volume 157 (2004) no. 2, pp. 275-328 | DOI | MR | Zbl
[26] Fonctions zêtas des variétés de Siegel de dimension trois, Formes automorphes. II. Le cas du groupe (Astérisque), Volume 302, Société Mathématique de France, 2005, pp. 1-66 | Numdam | MR | Zbl
[27] Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie, Math. Z., Volume 139 (1974), pp. 69-84 | DOI | MR | Zbl
[28] Geometry of the eigencurve at critical Eisenstein series of weight 2, J. Théor. Nombres Bordeaux, Volume 27 (2015) no. 1, pp. 183-197 | DOI | Numdam | MR | Zbl
[29] An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, 1997, pp. 243-311 | DOI | MR | Zbl
[30] Modular forms, Springer, 1989, x+335 pages (Translated from the Japanese by Yoshitaka Maeda) | DOI | MR
[31] An explicit computation of -stabilized vectors, J. Théor. Nombres Bordeaux, Volume 26 (2014) no. 2, pp. 531-558 | DOI | Numdam | MR | Zbl
[32] Galois representations attached to automorphic forms on over CM fields, Compos. Math., Volume 150 (2014) no. 4, pp. 523-567 | DOI | MR | Zbl
[33] On -adic height pairings, Séminaire de Théorie des Nombres, Paris, 1990–91 (Progress in Mathematics), Volume 108, Birkhäuser, 1993, pp. 127-202 | DOI | MR | Zbl
[34] Selmer complexes, Astérisque, 310, Société Mathématique de France, 2006, viii+559 pages | Numdam | MR
[35] Control theorem for Greenberg’s Selmer groups of Galois deformations, J. Number Theory, Volume 88 (2001) no. 1, pp. 59-85 | DOI | MR | Zbl
[36] Prolongement analytique sur les variétés de Siegel, Duke Math. J., Volume 157 (2011) no. 1, pp. 167-222 | DOI | MR | Zbl
[37] Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular -functions, J. Lond. Math. Soc., Volume 88 (2013) no. 1, pp. 251-270 | DOI | MR | Zbl
[38] On classical Saito-Kurokawa liftings, J. Reine Angew. Math., Volume 604 (2007), pp. 211-236 | DOI | MR | Zbl
[39] Packet structure and paramodular forms, Trans. Am. Math. Soc., Volume 370 (2018) no. 5, pp. 3085-3112 | DOI | MR | Zbl
[40] Paramodular forms in CAP representations of , Acta Arith., Volume 194 (2020) no. 4, pp. 319-340 | DOI | MR | Zbl
[41] Sur les déformations -adiques de certaines représentations automorphes, J. Inst. Math. Jussieu, Volume 5 (2006) no. 4, pp. 629-698 | DOI | MR | Zbl
[42] The Iwasawa main conjectures for , Invent. Math., Volume 195 (2014) no. 1, pp. 1-277 | DOI | MR | Zbl
[43] Galois representations attached to Hilbert-Siegel modular forms, Doc. Math., Volume 15 (2010), pp. 623-670 | MR | Zbl
[44] On the -adic cohomology of Siegel threefolds, Invent. Math., Volume 114 (1993) no. 2, pp. 289-310 | DOI | MR | Zbl
[45] Several-variable -adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, Ann. Sci. Éc. Norm. Supér., Volume 32 (1999) no. 4, pp. 499-574 | DOI | Numdam | MR | Zbl
[46] Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J., Volume 106 (2001) no. 3, pp. 485-525 | DOI | MR | Zbl
[47] Sur les représentations -adiques associées aux représentations cuspidales de , Formes automorphes. II. Le cas du groupe (Astérisque), Volume 302, Société Mathématique de France, 2005, pp. 151-176 | MR | Zbl
[48] Four dimensional Galois representations, Formes automorphes. II. Le cas du groupe (Astérisque), Volume 302, Société Mathématique de France, 2005, pp. 67-150 | Numdam | MR | Zbl
[49] Geometric Euler systems for locally isotropic motives, Compos. Math., Volume 140 (2004) no. 2, pp. 317-332 | DOI | MR | Zbl
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