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Affinoids in the Lubin–Tate perfectoid space and special cases of the local Langlands correspondence

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Abstract

Following Weinstein, Boyarchenko–Weinstein and Imai–Tsushima, 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.

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Notes

  1. In this paper by an affinoid we mean an open affinoid adic subspace. Also we often refer to the special fiber of a formal scheme \(\mathscr {A}\) over \(\mathcal {O}_C\) simply as the reduction of \(\mathscr {A}\).

  2. In fact [11] only treats positive \(\nu \). However, \(\nu =0\) case is in a sense the simplest and seems to be well-known to experts. The Deligne–Lusztig variety appears in the reduction as in [30, 32]. See [28, §5] for an account in the perfectoid setting.

  3. As should be clear from the following discussion, the use of q-th power compatible systems of topologically nilpotent elements, or equivalently valued points of \({\mathrm {Nil}}^{\flat }\), is crucial in this paper. Accordingly, we employ subtle notation related to \({\mathrm {Nil}}^{\flat }\). We do not explain such notation in this introduction and refer the reader to the main body of the paper, notably to (2.6)–(2.9) and Sect. 2.3.

  4. Strictly speaking, \(\Delta (\varvec{X}_1, \dots , \varvec{X}_n)\) denotes the system \((\Delta ^{q^{-l}}(\varvec{X}_1, \dots , \varvec{X}_n))_{l\ge 0}\) in (3.14).

  5. More precisely, we mean \(\xi \in \mathcal {M}_{H_0, \infty , \overline{\eta }}^{\mathrm{ad}}(C, \mathcal {O}_C)\) here. We use similar abbreviations.

  6. Here the inequality is written as if \(i<j\), but this is only for a notational convenience. We do not assume \(i<j\) and the argument clearly works without this assumption.

  7. In the assertions to follow, the relations between various \(y_i\) (resp. z) and various \(y_i'\) (resp. \(z'\)) are given in the proof.

  8. Since we are working with perfect rings, there are many other obvious possibilities; for instance, we may leave out all r from the definition of the above map.

  9. This follows either from the tame ramification assumption [21, Lemma 3], [8, Remarks after (1.4.10), (1.3.8)], or from Proposition 3.26.

  10. This follows either from the tame ramification assumption [1, (6.2.3)], or from Remark 3.28.

  11. Here, \(\nu =rn+s\). Complicated values like \(r-\nu +i-1-l\) result from our choices of the normalization (3.19) and the definition “\(w=\overline{(d^{-1}-1)\varphi _D^{-m/2}}\)” (instead of \(w=\overline{\varphi _D^{-m/2}(d^{-1}-1)}\)) in Proposition 3.29 (2).

  12. In [1, 4], this proposition is stated under the assumption that \(k=\mathbb {F}_p\). However, it plays a role only in the discussion of the computation of trace invariants and this proposition remains true without the assumption.

  13. Although this proposition allows us to compute the cohomology of \(Z_{\nu }\) for any \(\nu \) not divisible by n, only the cases where n and \(\nu \) are coprime are relevant to Main Theorem; we find the result interesting nonetheless.

  14. In fact, it also asserts that the cohomology in degree m is pure of weight m. However, we only need the dimension assertions in what follows.

  15. Note that some authors further impose the triviality of \(\xi |_{U_F^{i+1}}\) in the definition of minimality. This definition is taken from [3, §2.2] (except that i is assumed to be positive there). They also discuss jumps of possibly not minimal pairs.

  16. Thus, we change notation here; in Sect. 4\(\psi \) generally denotes characters of k.

  17. The conclusion below is trivial if \(\nu \) is odd and thus \(Z=S_{1, \nu }\); one can take \(y=1, z=x\).

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Acknowledgements

This paper is an improved version of the author’s Ph.D. thesis submitted to The University of Tokyo in January 2016. He wishes to express his deepest gratitude to his supervisor Takeshi Tsuji for his patient guidance, constant encouragement and uplifting words. Next he is grateful to Yoichi Mieda for making many comments on the previous version of this paper and especially for providing an argument to extend the result to the mixed-characteristic setting. Moreover he wants to thank Naoki Imai, Tomoki Mihara and Takahiro Tsushima for inspiring discussions, comments and for answering questions. He also wants to thank Tetsushi Ito for helpful comments. Furthermore he thanks Takeshi Saito for his encouragement and support. Finally he thanks the referee for reading the paper very carefully, pointing out many misprints and inaccuracies and suggesting a number of improvements in expositions. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan; the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University; Iwanami Fujukai Foundation; and JSPS KAKENHI Grant Number 19K14503.

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  1. Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan

    Kazuki Tokimoto

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Communicated by Wei Zhang.

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Tokimoto, K. Affinoids in the Lubin–Tate perfectoid space and special cases of the local Langlands correspondence. Math. Ann. 377, 1339–1425 (2020). https://doi.org/10.1007/s00208-020-01995-6

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