×
Andreatta, Fabrizio; Iovita, Adrian; Stevens, Glenn

Overconvergent modular sheaves and modular forms for \(\mathrm{GL}_{2/F}\). (English) Zbl 1326.14051

Isr. J. Math. 201, Part A, 299-359 (2014).
Authors’ abstract: Given a totally real field \(F\) and a prime integer \(p\) which is unramified in \(F\), we construct \(p\)-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of \(p\)-adic Hilbert modular forms. For \(F = \mathbb{Q}\), we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. F. Coleman [Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026)] and V. Pilloni [Ann. Inst. Fourier 63, No. 1, 219–239 (2013; Zbl 1316.11034)].
Reviewer: Gerd Faltings (Bonn)

Cited in 1 Review
Cited in 20 Documents

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F85 \(p\)-adic theory, local fields

Citations:

Zbl 0918.11026; Zbl 1316.11034
Cite Review PDF
Full Text: DOI

References:

[1] F. Andreatta and C. Gasbarri, The canonical subgroup for families of abelian varieties, Compositio Mathematica 143 (2007), 566-602. · Zbl 1219.11087
[2] F. Andreatta and E. Goren, Hilbert modular forms: mod p and p-Adic Aspects, American Mathematical Society Memoirs 819 (2005). · Zbl 1071.11023
[3] F. Andreatta, A. Iovita and V. Pilloni, p-Adic families of Siegel cuspforms, Annals of Mathematics, to appear. · Zbl 1394.11045
[4] F. Andreatta, A. Iovita and V. Pilloni, The cuspidal part of the Hilbert modular eigenvariety, (2012), submitted. · Zbl 1417.11063
[5] O. Brinon, Représentations p-adiques cristallines et de de Rham dans le cas relatif, Mémoires de la Société Mathématique de France 112 (2008). · Zbl 1170.14016
[6] C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces, Annals of Mathematics 131 (1990), 541-554. · Zbl 0754.14030 · doi:10.2307/1971469
[7] R. Coleman, p-Adic Banach spaces and families of modular forms, Inventiones Mathematicae 127 (1997), 417-479. · Zbl 0918.11026 · doi:10.1007/s002220050127
[8] R. Coleman, The canonical subgroup of E is Spec \[R[x]/({x^p} + \frac{p}{{{E_{p - 1}}(E,\omega )}}x)\], The Asian Journal of Mathematics 9 (2005), 257-260. · Zbl 1101.14058 · doi:10.4310/AJM.2005.v9.n2.a10
[9] R. Coleman, Classical and overconvergent modular forms, Inventiones Mathematicae 124 (1996), 215-241. · Zbl 0851.11030 · doi:10.1007/s002220050051
[10] G. Faltings and C.-L. Chai, Degeneration of abelian varieties. With an appendix by David Mumford, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 22 (1990). · Zbl 0744.14031
[11] G. Faltings, Hodge-Tate structures of modular forms, Mathematische Annalen 278 (1987), 133-149. · Zbl 0646.14026 · doi:10.1007/BF01458064
[12] G. Faltings, Almost étale extensions, in Cohomologie p-adiques et applications arithmétiques, II, Astérisque 279 (2002). · Zbl 1027.14011
[13] L. Fargues, L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld et applications cohomologiques, in L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld, Progress in Mathematics 262 (2008), 1-325. · Zbl 1136.14001
[14] L. Fargues, Le sous-group canonique des groupes de Barsotti-Tate tronqués déchelon quelconque, With collaboration of Yichao Tian, Annales Scientifiques de lÉcole Normale Supérieure 44 (2011), 905-961. · Zbl 1331.14044
[15] O. Gabber and L. Ramero, Foundations of almost ring theory, 2011, preprint. · Zbl 1045.13002
[16] E. Z Goren and Payman L. Kassaei, Canonical subgroups over Hilbert modular varieties, Journal für die Reine und Angewandte Mathematik 670 (2012), 1-63. · Zbl 1264.14038 · doi:10.1515/CRELLE.2011.149
[17] Grothendieck, A., Revêtements étales et groupe fondamental, No. 224 (1971), Berlin
[18] H. Hida, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Inventiones Mathematicae 85 (1986), 545-613. · Zbl 0612.10021 · doi:10.1007/BF01390329
[19] Katz, N., p-Adic properties of modular forms and modular schemes, No. 350, 69-190 (1973), Berlin · Zbl 0271.10033
[20] N. Katz, p-Adic L-functions for CM fields, Inventiones Mathematicae 49 (1978), 199-297. · Zbl 0417.12003 · doi:10.1007/BF01390187
[21] M. Kisin and K.-F. Lai, Overconvergent Hilbert modular forms, American Journal of Mathematics 127 (2005), 735-783. · Zbl 1129.11020 · doi:10.1353/ajm.2005.0027
[22] A. Ogus, F-Isocrystals and de Rham cohomology II-convergent isocrystals, Duke Mathematical Journal 51 (1984), 765-849. · Zbl 0584.14008 · doi:10.1215/S0012-7094-84-05136-6
[23] V. Pilloni, Overconvergent modular forms, Annales de l’Institut Fourier (Grenoble) 63 (2013), 219-239. · Zbl 1316.11034 · doi:10.5802/aif.2759
[24] M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Mathematica 36 (1978), 255-335 · Zbl 0386.14006
[25] P. Valabrega, A few theorems on completion of excellent rings, Nagoya Mathematics Journal 61 (1976), 127-133. · Zbl 0319.13008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.