×

Affinoids in the Lubin-Tate perfectoid space and special cases of the local Langlands correspondence. (English) Zbl 1469.11150

Summary: Following J. Weinstein [Doc. Math. 15, 981–1007 (2010; Zbl 1211.14027)], M. Boyarchenko and J. Weinstein [J. Am. Math. Soc. 29, No. 1, 177–236 (2016; Zbl 1336.11074)] and N. Imai and T. Tsushima [Contemp. Math. 691, 157–180 (2017; Zbl 1434.11118)], we construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the cohomology of the reduction of each formal model realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for certain representations. In the terminology of the essentially tame local Langlands correspondence, the representations treated here are characterized as being parametrized by minimal admissible pairs in which the field extensions are totally ramified.

MSC:

11F77 Automorphic forms and their relations with perfectoid spaces
11F80 Galois representations
14G45 Perfectoid spaces and mixed characteristic
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bushnell, C.J., Fröhlich, A.: Gauss Sums and \(p\)-adic Division Algebras, Lecture Notes in Mathematics, vol. 987. Springer, Berlin (1983) · Zbl 0507.12008
[2] Bushnell, C.J., Henniart, G.: Local tame lifting for \(\text{GL} (n)\) II: wildly ramified supercuspidals. Astérisque 254 (1999) · Zbl 0920.11079
[3] Bushnell, CJ; Henniart, G., The essentially tame local Langlands correspondence. I, J. Am. Math. Soc., 18, 3, 685-710 (2005) · Zbl 1073.11070
[4] Bushnell, CJ; Henniart, G., The essentially tame local Langlands correspondence, II: totally ramified representations, Compos. Math., 141, 4, 979-1011 (2005) · Zbl 1137.11074
[5] Bushnell, C.J., Henniart, G.: The Local Langlands Conjecture for \(\text{ GL }(2)\), Grundlehren der Mathematischen Wissenschaften, vol. 335. Springer, Berlin (2006) · Zbl 1100.11041
[6] Bushnell, CJ; Henniart, G., The essentially tame local Langlands correspondence, III: the general case, Proc. Lond. Math. Soc. (3), 101, 2, 497-553 (2010) · Zbl 1198.22009
[7] Bushnell, C.J., Henniart, G.: The essentially tame Jacquet-Langlands correspondence for inner forms of \(\text{ GL }(n)\). Pure Appl. Math. Q. 7(3, Special Issue: In honor of Jacques Tits), 469-538 (2011) · Zbl 1244.11053
[8] Bushnell, C.J., Kutzko, P.C.: The Admissible Dual of \(\text{ GL } (N)\) via Compact Open Subgroups, Annals of Mathematics Studies, vol. 129. Princeton University Press, Princeton (1993) · Zbl 0787.22016
[9] Boyarchenko, M.: Deligne-Lusztig constructions for unipotent and \(p\)-adic groups (2012). arXiv:1207.5876
[10] Boyarchenko, M., Weinstein, J.: Geometric realization of special cases of local Langlands and Jacquet-Langlands correspondences (2013). arXiv:1303.5795
[11] Boyarchenko, M.; Weinstein, J., Maximal varieties and the local Langlands correspondence for \({GL}(n)\), J. Am. Math. Soc., 29, 1, 177-236 (2016) · Zbl 1336.11074
[12] Boyer, P., Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math., 138, 3, 573-629 (1999) · Zbl 1161.11408
[13] Bump, D.: Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1997) · Zbl 0911.11022
[14] Carayol, H., Représentations cuspidales du groupe linéaire, Ann. Sci. Éc. Norm. Sup., 17, 191-225 (1984) · Zbl 0549.22009
[15] Deligne, P.: Cohomologie étale. Lecture Notes in Mathematics, vol. 569. Springer, Berlin (1977). Séminaire de Géométrie Algébrique du Bois-Marie SGA \(4{\frac{1}{2}} \), Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier · Zbl 0345.00010
[16] Deligne, P., La conjecture de Weil II, Publ. Math. Inst. Hautes Études Sci., 52, 137-252 (1980) · Zbl 0456.14014
[17] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. Math. (2), 103, 1, 103-161 (1976) · Zbl 0336.20029
[18] Drinfeld, V.G.: Elliptic modules. Mat. Sb. (N.S.) 94(136), 594-627, 656 (1974)
[19] Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001). With an appendix by Vladimir G. Berkovich · Zbl 1036.11027
[20] Hedayatzadeh, S.M.H.: Exterior powers of Barsotti-Tate groups. Ph.D. thesis, ETH Zürich (2010)
[21] Howe, RE, Tamely ramified supercuspidal representations of \(Gl_n\), Pac. J. Math., 73, 2, 437-460 (1977) · Zbl 0404.22019
[22] Huber, R., Continuous valuations, Math. Z., 212, 3, 455-477 (1993) · Zbl 0788.13010
[23] Huber, R., A generalization of formal schemes and rigid analytic varieties, Math. Z., 217, 4, 513-551 (1994) · Zbl 0814.14024
[24] Imai, N., Tsushima, T.: Affinoids in Lubin-Tate surfaces with exponential full level two. In: Around Langlands correspondences, Contemp. Math., vol. 691, pp. 157-180. Amer. Math. Soc., Providence (2017) · Zbl 1434.11118
[25] Imai, N., Tsushima, T.: Affinoids in the Lubin-Tate perfectoid space and simple supercuspidal representations I: tame case. Int. Math. Res. Not. (2018). 10.1093/imrn/rny229 · Zbl 1469.11149
[26] Imai, N., Tsushima, T.: Affinoids in the Lubin-Tate perfectoid space and simple supercuspidal representations II: wild case (2018). arXiv:1603.04693v2
[27] Lickorish, R.: An Introduction to Knot Theory, Graduate Texts in Mathematics, vol. 175. Springer, Berlin (1997) · Zbl 0886.57001
[28] Mieda, Y.: Geometric approach to the explicit local Langlands correspondence (2016). arXiv:1605.00511 · Zbl 1310.22015
[29] Scholze, P., Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci., 116, 245-313 (2012) · Zbl 1263.14022
[30] Weinstein, J., Good reduction of affinoids on the Lubin-Tate tower, Doc. Math., 15, 981-1007 (2010) · Zbl 1211.14027
[31] Weinstein, J., Semistable models for modular curves of arbitrary level, Invent. Math., 205, 2, 459-526 (2016) · Zbl 1357.14034
[32] Yoshida, T.: On non-abelian Lubin-Tate theory via vanishing cycles. In: Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), Adv. Stud. Pure Math., vol. 58, pp. 361-402. Math. Soc. Japan, Tokyo (2010) · Zbl 1257.11103
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.