Overconvergent Hilbert modular forms via perfectoid modular varieties
Annales de l'Institut Fourier, Online first, 86 p.

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary our definitions in p-adic families. We show that the resulting spaces are isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and Pilloni. Finally, we give a new direct construction of sheaves of arithmetic Hilbert modular forms, and compare this to the construction via descent from the geometric case.

Nous donnons une nouvelle construction de formes modulaires de Hilbert p-adiques surconvergentes en utilisant les variétés perfectoïdes de Shimura de Scholze au niveau infini et l’application de périodes de Hodge–Tate. La définition est analytique, ressemblant étroitement à celle des formes modulaires de Hilbert complexes en tant que fonctions holomorphes satisfaisant une propriété de transformation sous des sous-groupes de congruence. Comme cas particulier, nous revisitons d’abord le cas des formes modulaires elliptiques, prolongeant les travaux récents de Chojecki, Hansen et Johansson. Nous construisons ensuite des faisceaux de formes modulaires géométriques de Hilbert, ainsi que des sous-faisceaux de formes modulaires entières, et modifions nos définitions en familles p-adiques. Nous montrons que les espaces résultants sont isomorphes comme modules de Hecke à ceux construits par Andreatta, Iovita et Pilloni. Enfin, nous donnons une nouvelle construction directe de faisceaux de formes modulaires arithmétiques de Hilbert, et la comparons à la construction par descente à partir du cas géométrique.

Online First:
DOI: 10.5802/aif.3560
Classification: 11F77, 11F41, 11F33
Keywords: Hilbert, perfectoid, overconvergent.
Keywords: Hilbert, perfectoïde, surconvergence.
Birkbeck, Christopher 1; Heuer, Ben 2; Williams, Chris 3

1 Department of Mathematics University College London Gower street, London WC1E 6BT (UK)
2 Mathematical Institute University of Bonn Endenicher Allee 60, 53012 Bonn (Germany)
3 Mathematics Institute University of Warwick Coventry CV4 7AL (UK)
     author = {Birkbeck, Christopher and Heuer, Ben and Williams, Chris},
     title = {Overconvergent {Hilbert} modular forms via perfectoid modular varieties},
     journal = {Annales de l'Institut Fourier},
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     year = {2023},
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Birkbeck, Christopher; Heuer, Ben; Williams, Chris. Overconvergent Hilbert modular forms via perfectoid modular varieties. Annales de l'Institut Fourier, Online first, 86 p.

[1] Andreatta, F.; Iovita, A.; Pilloni, V. p-adic families of Siegel modular cusp forms, Ann. of Math. (2), Volume 181 (2015) no. 2, pp. 623-697 | DOI | MR | Zbl

[2] Andreatta, F.; Iovita, A.; Pilloni, V. The adic, cuspidal, Hilbert eigenvarieties, Res. Math. Sci., Volume 3 (2016), 34, 36 pages | DOI | MR | Zbl

[3] Andreatta, F.; Iovita, A.; Pilloni, V. On Overconvergent Hilbert Modular cusp forms, Arithmétique p-adique des formes de Hilbert (Astérisque), Volume 382, Société Mathématique de France (SMF), Paris, 2016, pp. 163-192 | Zbl

[4] Andreatta, F.; Iovita, A.; Pilloni, V. Le halo spectral, Ann. Sci. Éc. Norm. Supér. (4), Volume 51 (2018) no. 3, pp. 603-655 | DOI | MR | Zbl

[5] Bartenwerfer, W. Der erste Riemannsche Hebbarkeitssatz im nichtarchimedischen Fall, J. Reine Angew. Math., Volume 286(287) (1976), pp. 144-163 | DOI | MR | Zbl

[6] Buzzard, K. Eigenvarieties, L-functions and Galois representations (London Math. Soc. Lecture Note Ser.), Volume 320, Cambridge Univ. Press, Cambridge, 2007, pp. 59-120 | DOI | MR | Zbl

[7] Caraiani, A.; Scholze, P. On the generic part of the cohomology of compact unitary Shimura varieties, Ann. of Math. (2), Volume 186 (2017) no. 3, pp. 649-766 | DOI | MR | Zbl

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

[9] Chojecki, P.; Hansen, D.; Johansson, C. Overconvergent modular forms and perfectoid Shimura curves, Doc. Math., Volume 22 (2017), pp. 191-262 | MR | Zbl

[10] Hansen, D. Quotients of adic spaces by finite groups (to appear in Math. Res. Lett. http://www.davidrenshawhansen.com/adicgpquotient.pdf)

[11] Hansen, D.; Johansson, C. Perfectoid Shimura varieties and the Calegari–Emerton conjectures (preprint https://arxiv.org/abs/2011.03951)

[12] Hansen, D.; Kedlaya, K. S. Sheafiness criteria for Huber rings (preprint https://kskedlaya.org/papers/criteria.pdf)

[13] Heuer, B. Cusps and q-expansion principles for modular curves at infinite level (preprint https://arxiv.org/abs/2002.02488)

[14] Heuer, B. Line bundles on rigid spaces in the v-topology (preprint https://arxiv.org/abs/2012.07918)

[15] Heuer, B. Perfectoid geometry of p-adic modular forms, Ph. D. Thesis, King’s College London (2019)

[16] Hida, H. p-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004, xii+390 pages | DOI | MR | Zbl

[17] Howe, S. Overconvergent modular forms and the p-adic Jacquet-Langlands correspondence, Ph. D. Thesis, University of Chicago (2017) (https://knowledge.uchicago.edu/record/787)

[18] Huber, R. Étale cohomology of rigid analytic varieties and adic spaces, 30, Springer, 2013

[19] 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 Mathematics:), Volume 350, Springer, Berlin, 1973, pp. 69-190 | MR | Zbl

[20] Kedlaya, K. S.; Liu, R. Relative p-adic Hodge theory, II: Imperfect period rings (preprint https://arxiv.org/abs/1602.06899)

[21] Kisin, M.; Lai, K. F. Overconvergent Hilbert modular forms, Amer. J. Math., Volume 127 (2005) no. 4, pp. 735-783 | MR | Zbl

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

[23] Mumford, D. Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970, viii+242 pages | MR | Zbl

[24] Pilloni, V. Formes modulaires surconvergentes, Ann. Inst. Fourier (Grenoble), Volume 1 (2013) no. 63, pp. 219-239 | DOI | Zbl

[25] Rapoport, M. Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Math., Volume 36 (1978) no. 3, pp. 255-335 | MR | Zbl

[26] Scholze, P. Étale cohomology of diamonds (preprint https://arxiv.org/abs/1709.07343)

[27] Scholze, P. p-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi, Volume 1 (2013), e1, 77 pages | DOI | MR | Zbl

[28] Scholze, P. On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2), Volume 182 (2015) no. 3, pp. 945-1066 | DOI | MR | Zbl

[29] Scholze, P.; Weinstein, J. p-adic geometry, UC Berkeley course notes, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ

[30] Scholze, P.; Weinstein, J. Moduli of p-divisible groups, Camb. J. Math., Volume 1 (2013) no. 2, pp. 145-237 | DOI | MR | Zbl

[31] Shen, X. Perfectoid Shimura varieties of abelian type, Int. Math. Res. Not. (2017) no. 21, pp. 6599-6653 | DOI | MR | Zbl

[32] Tian, Y.; Xiao, L. p-adic cohomology and classicality of overconvergent Hilbert modular forms, Arithmétique p-adique des formes de Hilbert (Astérisque), Société Mathématique de France (SMF), Paris, 2016 no. 382, pp. 73-162 | MR | Zbl

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