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The Local Langlands Correspondence for GL n over p-adic fields

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Abstract

We extend our methods from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL n over p-adic fields as well as the existence of ℓ-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001).

In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497–544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) describing the inertia-invariant nearby cycles in certain regular situations.

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Notes

  1. This is the Weil group representation underlying the Weil-Deligne representation attached to π. We ignore the monodromy operator in this article.

  2. It would be more customary to write δ instead of β, but following this convention would result in too many different δ’s throughout this article.

  3. We caution the reader that we use the language used e.g. in SGA; in Berkovich’s language, these sheaves would be called vanishing cycle sheaves.

  4. More precisely, one should say arguments involving Shimura varieties. Many statements from local harmonic analysis that are used in the local arguments, e.g. base-change of representations, are only proved by global means.

  5. Our terminology is the one used e.g. in SGA and differs from Berkovich’s terminology, where these sheaves are called vanishing cycle sheaves.

  6. With the convention on the signature explained in the next few lines.

  7. Note that the number r is denoted j by Kottwitz.

  8. At least in the case F=ℚ p , but the description generalizes without problems. Note that one may use the algebraizations constructed in Theorem 2.4 instead of the Shimura varieties considered in [23].

  9. A word of warning may be in order: One can send an irreducible smooth representation π to its supercuspidal support, considered as an element of \(\mathcal{A}_{F}\), and hence define L(π,s), but this does not agree with the usual definition in general.

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Acknowledgements

First of all, I thank my advisor M. Rapoport for explaining me the Langlands-Kottwitz method of counting points, which plays a crucial role in this article, for his encouragement to work on this topic, and for the many other things he taught me. Furthermore, my thanks go to Guy Henniart and Vincent Sécherre for their advice in type theory, among other things. Moreover, I am grateful for the financial support of the Hausdorff Center for Mathematics in Bonn, and the hospitality of the Institut Henri Poincaré and Harvard University, where part of this work was carried out.

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  1. Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Peter Scholze

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  1. Peter Scholze
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Correspondence to Peter Scholze.

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Scholze, P. The Local Langlands Correspondence for GL n over p-adic fields. Invent. math. 192, 663–715 (2013). https://doi.org/10.1007/s00222-012-0420-5

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