Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees

TIAN, Yichao

Rendiconti del Seminario Matematico della Università di Padova, Tome 132 (2014), pp. 133-230.

[Ab11] A. Abbes, Élments de Géométrie Rigide: Volume I. Construction et Étude Géométrique des Espaces Rigides, Prog. in Math. 286, Birkhäuser, Basel, 2010. | MR | Zbl

[AM04] A. Abbes and A. Mokrane, Sous-groupes canoniques et cycles évanescents $p$ -adiques pour les variééts abéliennes, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 117–162. | Numdam | MR | Zbl

[AG07] F. Andreatta and C. Gasbarri, The canonical subgroup for families of abelian varieties, Compos. Math. 143 (2007), 566–602. | MR | Zbl

[Ba78] W. Bartenwerfer, Die erst "metrische" Kohomologiegruppe glatter affinoider Räume, Nederl. Akad. Wetensch. Proc. Ser. A 40, no. 1, (1978), 114. | MR | Zbl

[Be96] P. Berthelot, Cohomologie rigide et cohomologie rigide à support propre, Première partie, prepublication 96-03, 1996, available at http://perso.univ-rennes1.fr/pierre.berthelot/.

[BBM82] P. Berthelot, L. Breen and W. Messing, Théorie de Dieudonné Cristalline II, LNM 930, Springer-Verlag, Berlin, 1982. | MR | Zbl

[BR93] D. Blasius and J. Rogawski, Motives for Hilbert modular forms, Invent. Math. 114 (1993), 55–87. | MR | Zbl

[Br99] C. Breuil, Une application du corps des normes, Compositio Math. 117 (1999), 189–203. | MR | Zbl

[Bu03] K. Buzzard, Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc. 16 (2003), no. 1, 29–55. | MR | Zbl

[Ca86] H. Carayol, Sur les représentations $\ell$ -adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Sup. (4) 19 (1986), 409–468. | Numdam | MR | Zbl

[Ch90] C. L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces, Ann. of Math. (2) 131, No. 3 (1990), 541–554. | MR | Zbl

[Col96] R. Coleman, Classical and overconvergent modular forms, Invent. Math. 124 (1996), no. 13, 215–241. | MR | Zbl

[Col97a] R. Coleman, Classical and overconvergent modular forms of higher level, J. Théor. Nombres Bordeaux 9 (1997), 395–403. | Numdam | MR | Zbl

[Col97b] R. Coleman, $p$ -adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479. | MR | Zbl

[CM98] R. Coleman and B. Mazur, The eigencurve, in: Galois representations in arithmetic algebraic geometry (Durham 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 1–113. | MR | Zbl

[DP94] P. Deligne and G. Pappas, Singularités des espaces de modules de Hilbert en les caractéristiques divisant le discriminant, Compositio Math. 90 (1994), 59–79. | Numdam | MR | Zbl

[DT04] M. Dimitrov and J. Tilouine, Variétés et formes modulaires de Hilbert arithmétiques pour ${\Gamma }_1(𝔠,𝔫)$ , in Geometric aspects of Dwork theory (A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser eds.), Walter de Gruyter Berlin, 2004, 555–614. | MR | Zbl

[Fa10] L. Fargues, La filtration de Harder-Narasimhan des schémas en groupes finis et plats, J. Reine Angew. Math. 645 (2010), 1–39. | MR | Zbl

[Fa09] L. Fargues (avec la collaboration de Y. Tian), La filtration canonique des points de torsion des groupes $p$ -divisibles, Ann. Sci. Éc. Norm. Sup. (4) 44 (2011), 905–961. | Numdam | MR | Zbl

[Go01] E. Goren, Lectures on Hilbert modular varieties and Hilbert modular forms, CRM Mono. Series vol. 14, American Math. Soc., Providence, RI, 2002. | MR | Zbl

[GK09] E. Goren and P. Kassaei, Canonical subgroups over Hilbert modular varieties, J. Reine Angew. Math. 670 (2012), 163. | MR | Zbl

[GO00] E. Goren and F. Oort, Stratifications of Hilbert modular varieties, J. Algebraic Geom. 9 (2000), no. 1, 111–154. | MR | Zbl

[EGAIV] A. Grothendieck, Éléments de Géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) IV: Étude locale des schémas et des morphismes de schémas, Second Partie, Publ. Math. Inst. Hautes Études 24 (1965), 5–231. | Numdam | MR | Zbl

[SGA2] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Séminaire de Géométrie algébrique du Bois Marie 1962, Adv. Stu in Pure Math. vol. 2, North-Holland Publishing Co., Amsterdam, 1968. | MR | Zbl

[Ha10] S. Hattori, Canonical subgroups via Breuil-Kisin modules, Math. Z. 274 (2013), 933–953. | MR | Zbl

[Ks06] P. Kassaei, A gluing lemma and overconvergent modular forms, Duke Math. J. 132 (2006), 509–529. | MR | Zbl

[Ks09] P. Kassaei, Overconvergence and classicality: the case of curves, J. Reine Angew. Math. 631 (2009), 109–139. | MR | Zbl

[Ka73] N. Katz, $p$ -adic properties of modular schemes and modular forms, in: Lect. Notes in Math. 350 (1973), 69–170. | MR | Zbl

[Ka78] N. Katz, $p$ -adic $L$ -functions for CM-fields, Invent. Math. 49 (1978), 199–297. | MR | Zbl

[KL05] M. Kisin and K. Lai, Overconvergent Hilbert modular forms, Amer. J. Math. 127 (2005), 735–783. | MR | Zbl

[Ki06] M. Kisin, Crystalline representations and $F$ -crystals, in: Algebraic Geometry and Number Theory, in honor of Vladimir Drinfeld's 50th Birthday, V. Ginzburg editor, Progress in Math. 253, Birkhäuser Boston, 2006. | MR | Zbl

[Ki09a] M. Kisin, Moduli of finite flat group schemes and modularity, Ann. of Math. 170 (3), (2009), 1085–1180. | MR | Zbl

[Ki09b] M. Kisin, Modularity of 2-adic Barsotti-Tate representations, Invent. Math. 178 (3) (2009), 587–634. | MR | Zbl

[La10] E. Lau, A relation between Dieudonné displays and crystalline Dieudonné theory, preprint, arXiv:1006.2720v2. | MR | Zbl

[Li10] T. Liu, The correspondence between Barsotti-Tate groups and Kisin modules when $p=2$ , to appear in J. Théor. Nombres Bordeaux. | Numdam | MR | Zbl

[Ma86] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986. | MR | Zbl

[Pi09] V. Pilloni, Prolongement analytique sur les varités de Siegel, Duke Math. J. 157 (2011) No. 1, 167–222. | MR | Zbl

[PS11a] V. Pilloni and B. Stroh, Surconvergence et classicité : le cas déployé, preprint in 2011, available at the author's homepage.

[PS11b] V. Pilloni and B. Stroh, Surconvergence et classicité: le cas Hilbert, preprint in 2011, available at the author's homepage.

[Ra09] J. Rabinoff, Higher level canonical subgroups for $p$ -divisible groups, J. Inst. Math. Jussieu, 11 (2012), no. 2, 363–419. | MR | Zbl

[Ra74] M. Raynaud, Schémas en groupes de type $(p,...,p)$ , Bull. Soc. Math. France 102 (1974), 241–280. | Numdam | MR | Zbl

[Rap78] M. Rapoport, Compactification de l'espace de modules de Hilbert-Blumenthal, Compositio Math. 36 (1978), 255–335. | Numdam | MR | Zbl

[Sa10] S. Sasaki, Analytic continuation of overconvergent Hilbert eigenforms in the totally split case, Compositio Math. 146 (2010), no. 3, 541560. | MR | Zbl

[St10a] B. Stroh, Compactification de variétés de Siegel aux places de mauvaise réduction, Bull. Soc. Math. France 138 (2010), no. 2, 259–315. | Numdam | MR | Zbl

[St10b] B. Stroh, Compactification minimale et mauvaise réduction, Ann. Inst. Fourier 60 (2010), no. 3, p. 1035-1055. | Numdam | MR | Zbl

[Ta89] R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math., 98 (1989), 265–280. | MR | Zbl

[Ti10] Y. Tian, Canonical subgroups of Barsotti-Tate groups, Ann. of Math. 172, No. 2 (2010), 955–988. | MR | Zbl

[St97] H. Stamm, On the reduction of the Hilbert-Blumenthal moduli scheme with ${\Gamma }_0\left(p\right)$ -level structure, Forum Math. 9 (1997), no. 4, 405–455. | MR | Zbl

[Wi83] J.-P. Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux, Ann. Sci. Éc. Norm. Sup. (4) 16 (1983), 59–79. | Numdam | MR | Zbl

[Zi01] T. Zink, Windows for displays of $p$ -divisible groups, in Moduli of Abelian Varieties, Prog. in Math. 195, Birkhäuser, Basel, 2001, 491–518. | MR | Zbl