# zbMATH — the first resource for mathematics

Rigidity and a Riemann-Hilbert correspondence for $$p$$-adic local systems. (English) Zbl 1375.14090
The subject of this paper is relative $$p$$-adic Hodge theory. The main result (theorem 1.4) is the construction of a functor from the category of $$p$$-adic étale local systems on a smooth rigid analytic variety over a $$p$$-adic field, to the category of vector bundles with an integrable connection on the “base change to $$\mathrm{B}_{dR}$$” of the variety. This construction can be seen as a kind of Riemann-Hilbert correspondence.
A consequence of this construction is the fact (Theorem 1.1) that if the stalk of such a local system at one point is a de Rham $$p$$-adic Galois representation, then the stalk at every point is de Rham. In particular, geometric families of $$p$$-adic Galois representations are much more rigid than their arithmetic counterparts. A corollary of this rigidity result (Theorem 1.2) is that some representations coming from Shimura data are geometric in the sense of Fontaine-Mazur. The paper also contains (Theorem 1.6) an application of these constructions to the $$p$$-adic Simpson correspondence.
There is a detailed informative introduction that clearly explains the main results of the paper.

##### MSC:
 14G22 Rigid analytic geometry 14G35 Modular and Shimura varieties 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 11S15 Ramification and extension theory
Full Text:
##### References:
 [1] Abbes, A., Gros, M.: La suite spectrale de Hodge-Tate. arXiv:1509.03617 · Zbl 1330.14036 [2] Abbes, A., Gros, M., Tsuji, T.: The $$p$$-adic Simpson correspondence. Ann. Math. Stud. 193 (2016) · Zbl 1342.14045 [3] Bellovin, R, $$p$$ -adic Hodge theory in rigid analytic families, Algebra Number Theory, 9, 371-433, (2015) · Zbl 1330.14036 [4] Berger, L; Colmez, P, Familles de représentations de de Rham et monodromie p-adique, Astérisque No., 319, 303-337, (2008) · Zbl 1168.11020 [5] Berkovich, V.: Integration of one-forms on $$p$$ -adic analytic spaces. In: Annales of Mathematical Studies, vol. 162, p. 156. Princeton University Press, Princeton and Oxford (2006) · Zbl 1102.14022 [6] Bierstone, E; Milman, PD, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., 128, 207-302, (1997) · Zbl 0896.14006 [7] Brinon, O, Une généralisation de la théorie de Sen, Math. Ann., 327, 793-813, (2003) · Zbl 1072.11089 [8] Brinon, O.: Représentations $$p$$-adiques cristallines et de de Rham dans le cas relatif. Mémoire de la Soc. Math. France 112 (2008) · Zbl 1170.14016 [9] Conrad, B.: Lifting global representations with local properties. http://math.stanford.edu/ conrad/papers/locchar. (preprint) · Zbl 0597.20038 [10] Jong, AJ; Put, M, Étale cohomology of rigid spaces, Documenta Mathematica (Electron. J.), 1, 1-56, (1996) · Zbl 0922.14012 [11] Deligne, P., Milne, J.S.: Hodge cycles on abelian varieties. In: Hodge Cycles, Motives, and Shimura Varieties, LNM 900. Springer, Berlin (1982) · Zbl 0537.14006 [12] Faltings, G, A $$p$$-adic simpson correspondence, Adv. Math., 198, 847-862, (2005) · Zbl 1102.14022 [13] Henniart, G.: Représentations $$ℓ$$-adiques abéliennes. In: Seminar on Number Theory, Paris 1980-1981 (Paris, 1980/1981). Progr. Math., vol. 22, pp. 107-126. Birkhäuser, Boston (1982) · Zbl 1204.14010 [14] Hyodo, O, On variation of Hodge-Tate structures, Math. Ann., 284, 7-22, (1989) · Zbl 0645.14002 [15] Kedlaya, K, Good formal structures for flat meromorphic connections, I: surfaces, Duke Math. J., 154, 343-418, (2010) · Zbl 1204.14010 [16] Kedlaya, K.: Some slope theory for multivariate Robba rings. arXiv:1311.7468 · Zbl 1376.11029 [17] Kedlaya, K., Liu, R.: Relative $$p$$-adic Hodge theory: foundations, Astérisque No. 371 (2015) · Zbl 1370.14025 [18] Kedlaya, K., Liu, R.: Relative $$p$$-adic Hodge theory, II: imperfect period rings (2016). arXiv:1602.06899v1 · Zbl 0645.14002 [19] Kisin, M, Local constancy in p-adic families of Galois representations, Math. Z., 230, 569-593, (1999) · Zbl 0932.32028 [20] Kottwitz, R, Isocrystals with additional structure, Compos. Math., 56, 201-220, (1985) · Zbl 0597.20038 [21] Liu, R, Triangulation of refined families, Comment. Math Helvetici., 90, 831-904, (2015) · Zbl 1376.11029 [22] Milne, J.: Canonical models of (mixed) shimura varieties and automorphic vector bundles. Automorphic Forms, Shimura Varieties, and L-Functions, vol. I, pp. 283-414. Ann Arbor, MI (1988). Perspect. Math., vol. 10. Academic Press, Boston (1990) · Zbl 0932.32028 [23] Milne, J.: Introduction to Shimura varieties. Harmonic analysis, the trace formula and shimura varieties. Clay Math. Proc. 4, 265-378 (2005) · Zbl 1148.14011 [24] Scholze, P, Perfectoid spaces, Publ. Math. de l’IHÉS, 116, 245-313, (2012) · Zbl 1263.14022 [25] Scholze, P, $$p$$ -adic Hodge theory for rigid-analytic varieties, Forum Math. Pi, 1, e1, (2013) · Zbl 1297.14023 [26] Scholze, P.: Erratum to [25]. http://www.math.uni-bonn.de/people/scholze/pAdicHodgeErratum · Zbl 07054388 [27] Scholze, P., Weinstein, J.: $$p$$-adic geometry. Lecture notes from course Math274 at UC Berkeley in Fall 2014. https://math.berkeley.edu/ jared/Math274/ScholzeLectures · Zbl 1168.11020 [28] Simpson, C.: Higgs bundles and local systems. Publ. Math. de l’I.H.É,S., tome 75, pp. 5-95 (1992) · Zbl 0814.32003 [29] Tsuji, T, Purity for Hodge-Tate representations, Math. Ann., 350, 829-866, (2011) · Zbl 1239.14014 [30] Verdier, J.L.: Fonctorialité de catégories de faisceaux. Théorie des topos et cohomologie étale de schémas (SGA 4), Tome 1. In: Lect. Notes in Math., vol. 269, pp. 265-298. Springer, Berlin (1972)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.