Skip to main content Accessibility help

Galois level and congruence ideal for $p$ -adic families of finite slope Siegel modular forms

  • Andrea Conti (a1)

We consider families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for $\operatorname{GL}_{2}$ , via a $p$ -adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the $p$ -adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for $\operatorname{GL}_{2}$ to an eigenvariety for $\operatorname{GSp}_{4}$ , while the remainder appear as isolated points on the eigenvariety.

Hide All

Current address: Computational Arithmetic Geometry – IWR – Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

The author was supported by the Programs ArShiFo ANR-10-BLAN-0114 and PerCoLaTor ANR-14-CE25-0002-01.

Hide All
[AIP15] Andreatta, F., Iovita, A. and Pilloni, V., p-adic families of Siegel modular cuspforms , Ann. of Math. (2) 181 (2015), 623697.10.4007/annals.2015.181.2.5
[And87] Andrianov, A. N., Quadratic Forms and Hecke Operators, Grundlehren der Mathematischen Wissenschaften, vol. 286 (Springer, Berlin, 1987).
[And09] Andrianov, A. N., Twisting of Siegel modular forms with characters, and L-functions , St. Petersburg Math. J. 20 (2009), 851871.10.1090/S1061-0022-09-01076-0
[Bel12] Bellaïche, J., Eigenvarieties and adjoint  $p$ -adic  $L$ -functions, Preprint (2012).
[BC09] Bellaïche, J. and Chenevier, G., Families of Galois representations and Selmer groups, Astérisque, vol. 324 (Société Mathématique de France, Paris, 2009).
[Ber02] Berger, L., Représentations p-adiques et équations différentielles , Invent. Math. 148 (2002), 219284.
[Ber11] Berger, L., Trianguline representations , Bull. Lond. Math. Soc. 43 (2011), 619635.10.1112/blms/bdr036
[BC10] Berger, L. and Chenevier, G., Représentations potentiellement triangulines de dimension 2 , J. Théor. Nombres Bordeaux 22 (2010), 557574.
[BPS16] Bijakowski, S., Pilloni, V. and Stroh, B., Classicité de formes modulaires surconvergentes , Ann. of Math. (2) 183 (2016), 9751014.
[BGR84] Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261 (Springer, Berlin, 1984).10.1007/978-3-642-52229-1
[BR16] Brasca, R. and Rosso, G., Eigenvarieties for non-cuspidal modular forms over certain PEL Shimura varieties, Preprint (2016), arXiv:1605.05065.
[Buz07] Buzzard, K., Eigenvarieties , in L-functions and Galois representations, London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), 59120.10.1017/CBO9780511721267.004
[Car94] Carayol, H., Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet , Contemp. Math. 165 (1994), 213237.10.1090/conm/165/01601
[Che04] Chenevier, G., Familles p-adiques de formes automorphes pour GL(n) , J. reine angew. Math. 570 (2004), 143217.
[Che05] Chenevier, G., Une correspondance de Jacquet–Langlands p-adique , Duke Math. J. 126 (2005), 161194.
[Col96] Coleman, R., Classical and overconvergent modular forms , Invent. Math. 124 (1996), 214241.10.1007/s002220050051
[CM98] Coleman, R. and Mazur, B., The eigencurve , in Galois representations in arithmetic algebraic geometry, London Mathematical Society Lecture Note Series, vol. 254 (Cambridge University Press, Cambridge, 1998), 1113.
[Col08] Colmez, P., Représentations triangulines de dimension 2 , in Représentations p-adiques de groupes p-adiques I : représentations galoisiennes et (𝜑, 𝛤)-modules, Astérisque, vol. 319 (Société Mathématique de France, Paris, 2008), 213258.
[Con99] Conrad, B., Irreducible components of rigid analytic spaces , Ann. Inst. Fourier (Grenoble) 49 (1999), 905919.
[Con16] Conti, A., Big Galois image for  $p$ -adic families of positive slope automorphic forms, PhD thesis, Université Paris 13 (2016).
[CIT16] Conti, A., Iovita, A. and Tilouine, J., Big image of Galois representations associated with finite slope p-adic families of modular forms , in Elliptic curves, modular forms and Iwasawa theory: In Honour of John H. Coates’ 70th Birthday, Cambridge, UK, March 2015, Proceedings in Mathematics & Statistics, vol. 188, eds Loeffler, D. and Zerbes, S. L. (Springer, New York, 2016), 87124.10.1007/978-3-319-45032-2_3
[DiM13a] Di Matteo, G., On admissible tensor products in p-adic Hodge theory , Compos. Math. 149 (2013), 417429.
[DiM13b] Di Matteo, G., On triangulable tensor products of  $B$ -pairs and trianguline representations, Preprint (2013).
[DG12] Dimitrov, M. and Ghate, E., On classical weight one forms in Hida families , J. Théorie Nombres Bordeaux 24 (2012), 669690.
[Eme14] Emerton, M., Local-global compatibility in the  $p$ -adic Langlands programme for  $\operatorname{GL}_{2/\mathbb{Q}}$ , Preprint (2014).
[Fal89] Faltings, G., Crystalline cohomology and Galois representations , in Algebraic analysis, geometry, and volume theory (Johns Hopkins University Press, Baltimore, MD, 1989), 2580.
[Fis02] Fischman, A., On the image of 𝜆-adic Galois representations , Ann. Inst. Fourier (Grenoble) 52 (2002), 351378.
[GT05] Genestier, A. and Tilouine, J., Systèmes de Taylor–Wiles pour GSp4 , in Formes automorphes II. Le cas du groupe GSp(4), Astérisque, vol. 302 (Société Mathématique de France, Paris, 2005), 177290.
[Han17] Hansen, D., Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality , J. Reine Angew. Math. 730 (2017), 164.10.1515/crelle-2014-0130
[Hid15] Hida, H., Big Galois representations and p-adic L-functions , Compos. Math. 151 (2015), 603654.10.1112/S0010437X14007684
[HT15] Hida, H. and Tilouine, J., Big image of Galois representations and congruence ideals , in Arithmetic and Geometry, eds Dieulefait, L., Faltings, G., Heath-Brown, D. R., Manin, Y. I., Moroz, B. Z. and Wintenberger, J.-P. (Cambridge University Press, Cambridge, 2015), 217254.
[deJ95] de Jong, A. J., Crystalline Dieudonné theory via formal and rigid geometry , Publ. Math. Inst. Hautes Études Sci. 82 (1995), 596.10.1007/BF02698637
[KPX14] Kedlaya, K. S., Pottharst, J. and Xiao, L., Cohomology of arithmetic families of (𝜑, 𝛤)-modules , J. Amer. Math. Soc. 27 (2014), 10431115.
[KS02] Kim, H. H. and Shahidi, F., Functorial products for GL2 × GL3 and the symmetric cube for GL2 , Ann. of Math. (2) 155 (2002), 837893.
[Kis03] Kisin, M., Overconvergent modular forms and the Fontaine–Mazur conjecture , Invent. Math. (2003), 363454.
[Lan16] Lang, J., On the image of the Galois representation associated to a non-CM Hida family , Algebra Number Theory 10 (2016), 155194.
[Liv89] Livné, R., On the conductors of modulo representations coming from modular forms , J. Number Theory 31 (1989), 133141.
[Lud14] Ludwig, J., p-adic functoriality for inner forms of unitary groups in three variables , Math. Res. Lett. 21 (2014), 141148.
[Maz89] Mazur, B., Deforming Galois representations , in Galois groups over ℚ (Springer, Berlin, 1989), 385437.
[Mom81] Momose, F., On the -adic representations attached to modular forms , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 89109.
[OMe78] O’Meara, O. T., Symplectic groups, Mathematical Surveys, vol. 16 (American Mathematical Society, Providence, RI, 1978).
[Pil12] Pilloni, V., Modularité, formes de Siegel et surfaces abéliennes , J. reine angew. Math. 666 (2012), 3582.
[Pin98] Pink, R., Compact subgroups of linear algebraic groups , J. Algebra 206 (1998), 438504.
[RS07] Ramakrishnan, D. and Shahidi, F., Siegel modular forms of genus 2 attached to elliptic curves , Math. Res. Lett. 14 (2007), 315332.
[Rib75] Ribet, K., On -adic representations attached to modular forms , Invent. Math. 28 (1975), 245276.
[Rib85] Ribet, K., On -adic representations attached to modular forms. II , Glasgow Math. J. 27 (1985), 185194.
[Rou96] Rouquier, R., Caractérisation des caractères et pseudo-caractères , J. Algebra 180 (1996), 571586.10.1006/jabr.1996.0083
[Sen73] Sen, S., Lie algebras of Galois groups arising from Hodge–Tate modules , Ann. of Math. (2) 97 (1973), 160170.
[Sen80] Sen, S., Continuous cohomology and p-adic Hodge theory , Invent. Math. 62 (1980), 89116.
[Sen93] Sen, S., An infinite dimensional Hodge–Tate theory , Bull. Soc. Math. France 121 (1993), 1334.10.24033/bsmf.2199
[Tay91] Taylor, R., Galois representations associated to Siegel modular forms of low weight , Duke Math. J. 63 (1991), 281332.
[Taz85] Tazhetdinov, S., Subnormal structure of symplectic groups over local rings , Math. Notes Acad. Sci. USSR 37 (1985), 164169.
[Urb05] Urban, E., Sur les représentations p-adiques associées aux représentations cuspidales de GSp4(Q) , in Formes automorphes II. Le cas du groupe GSp(4), Astérisque, vol. 302 (Société Mathématique de France, Paris, 2005), 151176.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed