An integral model of the perfectoid modular curve
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1193-1224.

We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the coherent cohomology of the integral perfectoid curve. Specializing to the structural sheaf, we can describe the dual of the completed cohomology as the inverse limit of the integral cusp forms of weight 2 and trace maps.

Nous construisons un modèle entier de la courbe modulaire perfectoïde. Avec cet objet nous montrons des résultats d’annulation de la cohomologie cohérente au niveau perfectoïde. Nous utilisons un théorème de dualité locale au niveau fini pour obtenir une dualité pour la cohomologie cohérente au niveau infini. Finalement, en considérant le faisceau structural, nous obtenons une description du dual de la cohomologie complétée en termes des formes modulaires cuspidales de poids 2 et des traces normalisées.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.170
Classification: 14G35,  14F17,  14G45
Keywords: Perfectoid modular curve, completed cohomology, coherent cohomology
Juan Esteban Rodríguez Camargo 1

1 UMPA UMR 5669 CNRS, ENS de Lyon 46 allée d’Italie, 69364 Lyon Cedex 07, France
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Juan Esteban Rodríguez Camargo. An integral model of the perfectoid modular curve. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1193-1224. doi : 10.5802/jep.170. https://jep.centre-mersenne.org/articles/10.5802/jep.170/

[Bha17] B. Bhatt - “Lecture notes for a class on perfectoid spaces”, 2017

[BMS18] B. Bhatt, M. Morrow & P. Scholze - “Integral p-adic Hodge theory”, Publ. Math. Inst. Hautes Études Sci. 128 (2018), p. 219-397 | Article | MR: 3905467 | Zbl: 1446.14011

[BS14] B. Bhatt & P. Scholze - “The pro-étale topology for schemes”, 2014 | 1309.1198

[DR73] P. Deligne & M. Rapoport - “Les schémas de modules de courbes elliptiques”, in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lect. Notes in Math., vol. 349, Springer, Berlin, 1973, p. 143-316 | Zbl: 0281.14010

[Eme06] M. Emerton - “On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms”, Invent. Math. 164 (2006) no. 1, p. 1-84 | Article | MR: 2207783 | Zbl: 1090.22008

[Fon13] J.-M. Fontaine - “Perfectoïdes, presque pureté et monodromie-poids (d’après Peter Scholze)”, in Séminaire Bourbaki (2011/2012), Astérisque, vol. 352, Société Mathématique de France, Paris, 2013, p. 509-534, Exp. no. 1057 | Zbl: 1325.14033

[Har66] R. Hartshorne - Residues and duality, Lect. Notes in Math., vol. 20, Springer-Verlag, Berlin-New York, 1966 | MR: 222093 | Zbl: 0212.26101

[Heu19] B. Heuer - Perfectoid geometry of p-adic modular forms, Ph. D. Thesis, University of London, 2019

[Heu20] B. Heuer - “Cusps and q-expansion principles for modular curves at infinite level”, 2020 | 2002.02488

[Hub94] R. Huber - “A generalization of formal schemes and rigid analytic varieties”, Math. Z. 217 (1994) no. 4, p. 513-551 | Article | MR: 1306024 | Zbl: 0814.14024

[Hub96] R. Huber - Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math., vol. E30, Friedr. Vieweg & Sohn, Braunschweig, 1996 | Article | MR: 1734903 | Zbl: 0868.14010

[Kat81] N. M. Katz - “Serre-Tate local moduli”, in Algebraic surfaces (Orsay, 1976–78), Lect. Notes in Math., vol. 868, Springer, Berlin-New York, 1981, p. 138-202 | MR: 638600 | Zbl: 0477.14007

[KL19] K. S. Kedlaya & R. Liu - “Relative p-adic Hodge theory, II: Imperfect period rings”, 2019 | 1602.06899

[KM85] N. M. Katz & B. Mazur - Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985 | Article | MR: 772569 | Zbl: 0576.14026

[LT66] J. Lubin & J. Tate - “Formal moduli for one-parameter formal Lie groups”, Bull. Soc. math. France 94 (1966), p. 49-59 | Article | Numdam | MR: 238854 | Zbl: 0156.04105

[Lur20] J. Lurie - “Full level structures on elliptic curves” (2020), https://www.math.ias.edu/~lurie/papers/LevelStructures1.pdf

[Mor17] M. Morrow - “Foundations of perfectoid spaces”, Notes for some talks in the Fargues-Fontaine curve study group at Harvard, 2017

[Mum08] D. Mumford - Abelian varieties, TIFR Studies in Mathematics, vol. 5, Tata Institute of Fundamental Research, Bombay, 2008 | MR: 2514037 | Zbl: 1177.14001

[Sch12] P. Scholze - “Perfectoid spaces”, Publ. Math. Inst. Hautes Études Sci. 116 (2012), p. 245-313 | Article | Numdam | MR: 3090258 | Zbl: 1263.14022

[Sch13] P. Scholze - “p-adic Hodge theory for rigid-analytic varieties”, Forum Math. Pi 1 (2013), article ID e1, 77 pages | Article | MR: 3090230 | Zbl: 1297.14023

[Sch15] P. Scholze - “On torsion in the cohomology of locally symmetric varieties”, Ann. of Math. (2) 182 (2015) no. 3, p. 945-1066 | Article | MR: 3418533 | Zbl: 1345.14031

[SW13] P. Scholze & J. Weinstein - “Moduli of p-divisible groups”, Camb. J. Math. 1 (2013) no. 2, p. 145-237 | Article | MR: 3272049 | Zbl: 1349.14149

[SW20] P. Scholze & J. Weinstein - Berkeley lectures on p-adic geometry, Princeton University Press, 2020 | Zbl: 07178476

[Wei13] J. Weinstein - “Modular curves of infinite level”, Notes from the Arizona Winter School, 2013

[Wei16] J. Weinstein - “Semistable models for modular curves of arbitrary level”, Invent. Math. 205 (2016) no. 2, p. 459-526 | Article | MR: 3529120 | Zbl: 1357.14034

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