Locally analytic vectors and decompletion in mixed characteristic

Porat, Gal

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e173

Résumé

In p-adic Hodge theory and the p-adic Langlands program, Banach spaces with $\mathbf {Q}_p$-coefficients and p-adic Lie group actions are central. Studying the subrepresentation of G-locally analytic vectors, $W^{\mathrm {la}}$, is useful because $W^{\mathrm {la}}$ can be studied via the Lie algebra $\mathrm {Lie}(G)$, which simplifies the action of G. Additionally, $W^{\mathrm {la}}$ often behaves as a decompletion of W, making it closer to an algebraic or geometric object.This article introduces a notion of locally analytic vectors for W in a mixed characteristic setting, specifically for $\mathbf {Z}_p$-Tate algebras. This generalization encompasses the classical definition and also specializes to super-Hölder vectors in characteristic p. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic $0$ and characteristic p.Our main theorem shows that under certain conditions, the map $W \mapsto W^{\mathrm {la}}$ acts as a descent, and the derived locally analytic vectors $\mathrm {R}_{\mathrm {la}}^i(W)$ vanish for $i \geq 1$. This result extends Theorem C of [Por24], providing new tools for propagating information about locally analytic vectors from characteristic $0$ to characteristic p.We provide three applications: a new proof of Berger-Rozensztajn’s main result using characteristic $0$ methods, the introduction of an integral multivariable ring $\widetilde {\mathbf {A}}_{\mathrm {LT}}^{\dagger ,\mathrm {la}}$ in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring $\mathbf {{A}}_{\mathbf {Q}_p}$ from the theory of $(\varphi ,\Gamma )$-modules in terms of locally analytic vectors.

Détail

DOI : 10.1017/fms.2025.10121

Bibliographie

[Ami64] Amice, Y., ‘Interpolation p-adique’, Bull. Soc. Math. France 92 (1964), 117–180.10.24033/bsmf.1606 Google Scholar | DOI

[BC08] Berger, L. and Colmez, P., ‘Familles de représentations de de Rham et monodromie p-adique’, Astérisque 319 (2008), 303–337. Google Scholar

[BC16] Berger, L. and Colmez, P., ‘Théorie de Sen et vecteurs localement analytiques’, Ann. Sci. Éc. Norm. Supér. (4) 49(4) (2016), 947–970.10.24033/asens.2300 Google Scholar | DOI

[Ber14] Berger, L., ‘Lifting the field of norms’, J. Éc. polytech. Math. 1 (2014), 29–38.10.5802/jep.2 Google Scholar | DOI

[Ber16] Berger, L., ‘Multivariable ()-modules and locally analytic vectors’, Duke Math. J. 165(18) (2016), 3567–3595.10.1215/00127094-3674441 Google Scholar | DOI

[BL22] Bhatt, B. and Lurie, J., ‘Absolute prismatic cohomology’, preprint (2022), . Google Scholar | arXiv

[Boj74] Bojanic, R., ‘A simple proof of Mahler’s theorem on approximation of continuous functions of a p-adic variable by polynomials’, J. Number Theory 6 (1974), 412–415.10.1016/0022-314X(74)90038-9 Google Scholar | DOI

[BR22] Berger, L. and Rozensztajn, S., ‘Decompletion of cyclotomic perfectoid fields in positive characteristic’, Ann. Henri Lebesgue 5 (2022), 1261–1276.10.5802/ahl.150 Google Scholar | DOI

[BR24] Berger, L. and Rozensztajn, S., ‘Super-Hölder vectors and the field of norms’, Algebr. Number Theory 19(1) (2024), 195–211.10.2140/ant.2025.19.195 Google Scholar | DOI

[CC98] Cherbonnier, F. and Colmez, P., ‘Représentations p-adiques surconvergentes’, Invent. Math. 133(3) (1998), 581–611.10.1007/s002220050255 Google Scholar | DOI

[Col08] Colmez, P., ‘Espaces vectoriels de dimension finie et représentations de de Rham’, Astérisque 319 (2008), 117–186. Google Scholar

[Col10] Colmez, P., ‘Fonctions d’une variable p-adique’, Astérisque 330 (2010), 9–29. Google Scholar

[DDSMS03] Dixon, J. D., Du Sautoy, M. P. F., Mann, A. and Segal, D., ‘Analytic pro-p groups’, Cambridge Stud. Adv. Math. 61 (2003), Cambridge Univ. Press, Cambridge. Google Scholar

[Eme17] Emerton, M., ‘Locally Analytic Vectors in Representations of Locally p-adic Analytic Groups’, Mem. Amer. Math. Soc. 248 (2017), American Mathematical Society, Providence, RI. Google Scholar

[FX14] Fourquaux, L. and Xie, B., ‘Triangulable -analytic (, )-modules of rank 2’, Algebr. Number Theory 7(10) (2014), 2545–2592.10.2140/ant.2013.7.2545 Google Scholar | DOI

[Gul19] Gulotta, D. R., ‘Equidimensional adic eigenvarieties for groups with discrete series’, Algebr. Number Theory 13(8) (2019), 1907–1940.10.2140/ant.2019.13.1907 Google Scholar | DOI

[JN19] Johansson, C. and Newton, J., ‘Extended eigenvarieties for overconvergent cohomology’, Algebr. Number Theory 13(1) (2019), 93–158.10.2140/ant.2019.13.93 Google Scholar | DOI

[Ked16] Kedlaya, K. S., ‘The Hochschild–Serre property for some p-adic analytic group actions’, Ann. Math. Québec 40(1) (2016), 149–157.10.1007/s40316-015-0051-5 Google Scholar | DOI

[KL11] Kedlaya, K. S. and Liu, R., ‘Relative p-adic Hodge theory. I. Foundations’, Astérisque 371 (2015) Google Scholar

[Laz65] Lazard, M., ‘Groupes analytiques p-adiques’, Publ. Math. Inst. Hautes Études Sci. 26 (1965), 5–219. Google Scholar

[Pan22] Pan, L., ‘On locally analytic vectors of the completed cohomology of modular curves’, Forum Math. Pi 10 (2022), e7.10.1017/fmp.2022.1 Google Scholar | DOI

[Por22a] Porat, G., ‘Lubin–Tate theory and overconvergent Hilbert modular forms of low weight’, Israel J. Math. 249(1) (2022), 431–476.10.1007/s11856-022-2317-3 Google Scholar | DOI

[Por22b] Porat, G., ‘Overconvergence of étale -modules in families’, preprint (2022), . Google Scholar | arXiv

[Por24] Porat, G., ‘Locally analytic vector bundles on the Fargues–Fontaine curve’, Algebr. Number Theory 18(5) (2024), 899–946.10.2140/ant.2024.18.899 Google Scholar | DOI

[Poy22] Poyeton, L., ‘Formal groups and lifts of the field of norms’, Algebr. Number Theory 16(2) (2022), 261–290.10.2140/ant.2022.16.261 Google Scholar | DOI

[RJRC22] Jacinto, J. Rodrigues and Camargo, J. Rodríguez, ‘Solid locally analytic representations of p-adic Lie groups’, Represent. Theory 26 (2022), 962–1024.10.1090/ert/615 Google Scholar | DOI

[SW20] Scholze, P. and Weinstein, J., ‘Berkeley Lectures on p-adic Geometry’, AMS (2020), Providence, RI. Google Scholar

[Tat67] Tate, J. T., ‘p-divisible groups’, Proc. Conf. Local Fields (1967), 158–183.10.1007/978-3-642-87942-5_12 Google Scholar | DOI

[Wan23] Wang, Y., ‘A p-adic Simpson correspondence for rigid analytic varieties’, Algebr. Number Theory 17(8) (2023), 1453–1499.10.2140/ant.2023.17.1453 Google Scholar | DOI