Le halo spectral

Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 603-655.

We construct a space of overconvergent modular forms of characteristic $p$, a compact operator on this space and we show that the characteristic series of this operator is the reduction modulo $p$ of Coleman's universal series. We prove that finite slope overconvergent modular forms in positive characteristic can be deformed to characteristic 0.

Nous construisons un espace de formes surconvergentes en caractéristique $p$, un opérateur compact sur cet espace et nous montrons que la série caractéristique de cet opérateur est la réduction modulo $p$ de la série universelle de Coleman. Nous démontrons que les formes surconvergentes en caractéristique $p$ de pente finie se déforment vers la caractéristique 0.

Andreatta, F.; Gasbarri, C. The canonical subgroup for families of abelian varieties, Compos. Math., Volume 143 (2007), pp. 566-602 (ISSN: 0010-437X) | DOI | MR | Zbl

Andreatta, F.; Iovita, A.; Stevens, G. Overconvergent modular sheaves and modular forms for ${\mathrm{𝐆𝐋}}_{2/F}$ , Israel J. Math., Volume 201 (2014), pp. 299-359 (ISSN: 0021-2172) | DOI | MR | Zbl

Abbès, A.; Mokrane, A. Sous-groupes canoniques et cycles évanescents $p$-adiques pour les variétés abéliennes, Publ. Math. IHÉS, Volume 99 (2004), pp. 117-162 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Berger, L.; Colmez, P. Familles de représentations de de Rham et monodromie $p$-adique, Astérisque, Volume 319 (2008), pp. 303-337 (ISBN: 978-2-85629-256-3, ISSN: 0303-1179) | Numdam | MR | Zbl

Berthelot, P. Cohomologie rigide et cohomologie rigide à supports propres (1996) (première partie, prépublication 96-03 https://perso.univ-rennes1.fr/pierre.berthelot/publis/Cohomologie_Rigide_I.pdf )

Buzzard, K.; Kilford, L. J. P. The 2-adic eigencurve at the boundary of weight space, Compos. Math., Volume 141 (2005), pp. 605-619 (ISSN: 0010-437X) | DOI | MR | Zbl

Bergdall, J.; Pollack, R. Arithmetic properties of Fredholm series for $p$-adic modular forms, Proc. Lond. Math. Soc., Volume 113 (2016), pp. 419-444 (ISSN: 0024-6115) | DOI | MR | Zbl

Buzzard, K. Eigenvarieties, Proceedings of the LMS Durham conference on $L$-functions and Galois representations (2004) | Zbl

Coleman, R. F.; Mazur, B., Galois representations in arithmetic algebraic geometry (Durham, 1996) (London Math. Soc. Lecture Note Ser.), Volume 254, Cambridge Univ. Press, Cambridge, 1998, pp. 1-113 | DOI | MR | Zbl

Coleman, R. F. The Spectral Halo (2014) (notes)

Coleman, R. F. Classical and overconvergent modular forms, Invent. math., Volume 124 (1996), pp. 215-241 (ISSN: 0020-9910) | DOI | MR | Zbl

Coleman, R. F. $p$-adic Banach spaces and families of modular forms, Invent. math., Volume 127 (1997), pp. 417-479 (ISSN: 0020-9910) | DOI | MR | Zbl

Deligne, P.; Rapoport, M. Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Math.), Volume 349, Springer, Berlin (1973), pp. 143-316 | MR | Zbl

Fargues, L. La filtration de Harder-Narasimhan des schémas en groupes finis et plats, J. reine angew. Math., Volume 645 (2010), pp. 1-39 (ISSN: 0075-4102) | DOI | MR | Zbl

Fargues, L. La filtration canonique des points de torsion des groupes $p$-divisibles, Ann. Sci. Éc. Norm. Supér., Volume 44 (2011), pp. 905-961 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. IHÉS, Volume 24 (1965), 231 pages (ISSN: 0073-8301) | Numdam | MR | Zbl

Hartl, U.; Pink, R. Vector bundles with a Frobenius structure on the punctured unit disc, Compos. Math., Volume 140 (2004), pp. 689-716 (ISSN: 0010-437X) | DOI | MR | Zbl

Huber, R. Continuous valuations, Math. Z., Volume 212 (1993), pp. 455-477 (ISSN: 0025-5874) | DOI | MR | Zbl

Huber, R. A generalization of formal schemes and rigid analytic varieties, Math. Z., Volume 217 (1994), pp. 513-551 (ISSN: 0025-5874) | DOI | MR | Zbl

Huber, R., Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996, 450 pages (ISBN: 3-528-06794-2) | DOI | MR | Zbl

Katz, N. M. $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Math.), Volume 350, Springer (1973), pp. 69-190 | MR | Zbl

Lütkebohmert, W. Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie, Math. Z., Volume 139 (1974), pp. 69-84 (ISSN: 0025-5874) | DOI | MR | Zbl

Liu, R.; Wan, D.; Xiao, L. The eigencurve over the boundary of weight space, Duke Math. J., Volume 166 (2017), pp. 1739-1787 (ISSN: 0012-7094) | DOI | MR | Zbl

Messing, W., Lecture Notes in Math., 264, Springer, Berlin-New York, 1972, 190 pages | MR | Zbl

Pilloni, V. Overconvergent modular forms, Ann. Inst. Fourier (Grenoble), Volume 63 (2013), pp. 219-239 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Scholze, P. On torsion in the cohomology of locally symmetric varieties, Ann. of Math., Volume 182 (2015), pp. 945-1066 (ISSN: 0003-486X) | DOI | MR | Zbl

Serre, J.-P. Endomorphismes complètement continus des espaces de Banach $p$-adiques, Publ. Math. IHÉS, Volume 12 (1962), pp. 69-85 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Tate, J. T., Proc. Conf. Local Fields (Driebergen, 1966), Springer, Berlin, 1967, pp. 158-183 | DOI | MR | Zbl

Tian, Y. Canonical subgroups of Barsotti-Tate groups, Ann. of Math., Volume 172 (2010), pp. 955-988 (ISSN: 0003-486X) | DOI | MR | Zbl

Valabrega, P. A few theorems on completion of excellent rings, Nagoya Math. J., Volume 61 (1976), pp. 127-133 http://projecteuclid.org/euclid.nmj/1118795739 (ISSN: 0027-7630) | DOI | MR | Zbl

Wiles, A. On ordinary $\lambda$-adic representations associated to modular forms, Invent. math., Volume 94 (1988), pp. 529-573 (ISSN: 0020-9910) | DOI | MR | Zbl