## Classicity of overconvergent modular forms. (Classicité de formes modulaires surconvergentes.)(English. French summary)Zbl 1407.11078

Summary: We generalize R. F. Coleman’s [Invent. Math. 124, No. 1–3, 215–241 (1996; Zbl 0851.11030)] classicity criterion for overconvergent modular forms on modular curves to the case of some PEL Shimura variety of type (A) or (C) associated to a reductive group unramified over $$\mathbb{Q}_p$$. Our demonstration is inspired by the analytic continuation method of K. Buzzard [J. Am. Math. Soc. 16, No. 1, 29–55 (2003; Zbl 1076.11029)] and P. L. Kassaei [Duke Math. J. 132, No. 3, 509–529 (2006; Zbl 1112.11020)].

### MSC:

 11G18 Arithmetic aspects of modular and Shimura varieties 11F85 $$p$$-adic theory, local fields 11F60 Hecke-Petersson operators, differential operators (several variables)

### Citations:

Zbl 0851.11030; Zbl 1076.11029; Zbl 1112.11020
Full Text:

### References:

 [1] F. Andreatta, A. Iovita, and V. Pilloni, ”$$p$$-adic families of Siegel modular cuspforms,” Ann. of Math., vol. 181, iss. 2, pp. 623-697, 2015. · Zbl 1394.11045 [2] A. Abbes and A. Mokrane, ”Sous-groupes canoniques et cycles évanescents $$p$$-adiques pour les variétés abéliennes,” Publ. Math. Inst. Hautes Études Sci., vol. 99, pp. 117-162, 2004. · Zbl 1062.14057 [3] P. Berthelot, Cohomologie rigide et cohomologie rigide à support propres, 1996. [4] S. Bosch, ”Half a century of rigid analytic spaces,” Pure Appl. Math. Q., vol. 5, iss. 4, Special Issue: In honor of John Tate. Part 1, pp. 1435-1467, 2009. · Zbl 1256.32023 [5] S. Bosch and W. Lütkebohmert, ”Formal and rigid geometry. II. Flattening techniques,” Math. Ann., vol. 296, iss. 3, pp. 403-429, 1993. · Zbl 0808.14018 [6] K. Buzzard, ”Analytic continuation of overconvergent eigenforms,” J. Amer. Math. Soc., vol. 16, iss. 1, pp. 29-55, 2003. · Zbl 1076.11029 [7] R. F. Coleman, ”Classical and overconvergent modular forms,” Invent. Math., vol. 124, iss. 1-3, pp. 215-241, 1996. · Zbl 0851.11030 [8] L. Fargues, ”La filtration de Harder-Narasimhan des schémas en groupes finis et plats,” J. Reine Angew. Math., vol. 645, pp. 1-39, 2010. · Zbl 1199.14015 [9] U. Görtz, ”On the flatness of models of certain Shimura varieties of PEL-type,” Math. Ann., vol. 321, iss. 3, pp. 689-727, 2001. · Zbl 1073.14526 [10] U. Görtz, ”On the flatness of local models for the symplectic group,” Adv. Math., vol. 176, iss. 1, pp. 89-115, 2003. · Zbl 1073.14526 [11] X. He, ”Normality and Cohen-Macaulayness of local models of Shimura varieties,” Duke Math. J., vol. 162, iss. 13, pp. 2509-2523, 2013. · Zbl 1327.14121 [12] C. Johansson, ”Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C,” Math. Ann., vol. 357, iss. 1, pp. 51-88, 2013. · Zbl 1346.11033 [13] P. L. Kassaei, ”A gluing lemma and overconvergent modular forms,” Duke Math. J., vol. 132, iss. 3, pp. 509-529, 2006. · Zbl 1112.11020 [14] R. Kiehl, ”Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie,” Invent. Math., vol. 2, pp. 191-214, 1967. · Zbl 0202.20101 [15] R. E. Kottwitz, ”Points on some Shimura varieties over finite fields,” J. Amer. Math. Soc., vol. 5, iss. 2, pp. 373-444, 1992. · Zbl 0796.14014 [16] K. Lan, Arithmetic Compactifications of PEL-type Shimura Varieties, Princeton, NJ: Princeton Univ. Press, 2013, vol. 36. · Zbl 1284.14004 [17] K. Lan, Higher Koecher’s principle. · Zbl 1416.11097 [18] K. Lan, ”Compactifications of PEL-type Shimura varieties in ramified characteristics,” Forum Math. Sigma, vol. 4, p. 98, 2016. · Zbl 1338.11056 [19] K. Lan, Integral models of toroidal compactifications with projective cone decompositions, 2015. [20] V. Pilloni, ”Prolongement analytique sur les variétés de Siegel,” Duke Math. J., vol. 157, iss. 1, pp. 167-222, 2011. · Zbl 1315.11033 [21] V. Pilloni and B. Stroh, Surconvergence et classicité : le cas Hilbert, 2011. [22] S. Sasaki, ”Analytic continuation of overconvergent Hilbert eigenforms in the totally split case,” Compos. Math., vol. 146, iss. 3, pp. 541-560, 2010. · Zbl 1206.11058 [23] Revêtements Étales et Groupe Fondamental, Grothendieck, A., Ed., New York: Springer-Verlag, 1971, vol. 224. · Zbl 0234.14002 [24] B. Stroh, ”Compactification de variétés de Siegel aux places de mauvaise réduction,” Bull. Soc. Math. France, vol. 138, iss. 2, pp. 259-315, 2010. · Zbl 1203.14048 [25] Y. Tian, ”Classicality of overconvergent Hilbert modular forms: case of quadratic inert degree,” Rend. Semin. Mat. Univ. Padova, vol. 132, pp. 133-229, 2014. · Zbl 1311.11036 [26] Y. Tian and L. Xiao, $$p$$-adic cohomology and classicality of overconvergent Hilbert modular forms, 2013. · Zbl 1423.11091 [27] T. Wedhorn, ”Ordinariness in good reductions of Shimura varieties of PEL-type,” Ann. Sci. École Norm. Sup., vol. 32, iss. 5, pp. 575-618, 1999. · Zbl 0983.14024
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