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• W88548708 abstract "The present notes cover 50% of the material presented in a course given jointly with Guy Henniart during the special semester “Formes Automorphes”, held at the Institut Henri Poincare in Paris between February and June 2000, as well as a little more material I didn’t have time to present. The purpose of the course was to explain two proofs of the local Langlands conjecture for p-adic fields, due respectively to Richard Taylor and myself [HT], and to Henniart [He5]. My lectures were naturally concerned with [HT], the main burden of which is to construct a candidate for a local Langlands correspondence, and to prove that this putative correspondence is (nearly) compatible with the global correspondence realized on the cohomology of certain specific Shimura varieties. The techniques applied derive mainly from arithmetic algebraic geometry: we study the bad reduction of the Shimura varieties in question by interpreting them locally/infinitesimally as formal deformation spaces for p-divisible groups with additional structure of a kind already studied by Drinfel’d. This yields a stratification of the special fiber, with particularly nice properties, in terms of p-rank of the universal p-divisible group. The cohomology of the Shimura varieties is then calculated by means of vanishing cycles on the bad special fiber. Thanks to Berkovich’s work on etale cohomology of (rigid) analytic spaces, the vanishing cycles can be computed infinitesimally, which permits determination of their stalks in terms of certain universal representation spaces. An extension, to our situation of bad reduction, of the trace formula techniques perfected by Langlands and Kottwitz for calculating zeta functions of Shimura varieties at places of good reduction, provides the necessary compatibility of local and global correspondences. My goal in the course was to present a self-contained account of the main results of [HT]. In so doing, I chose to sacrifice the description of the global structure of the strata in the special fiber, and of the vanishing cycles sheaves on the strata," @default.
• W88548708 created "2016-06-24" @default.
• W88548708 creator A5005841572 @default.
• W88548708 date "2005-01-01" @default.
• W88548708 modified "2023-09-27" @default.
• W88548708 title "THE LOCAL LANGLANDS CORRESPONDENCE: NOTES OF (HALF) A COURSE AT THE IHP SPRING 2000" @default.
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