The Geometry and Cohomology of Some Simple Shimura Varieties. (Am-151), Volume 151

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The book addresses significant advancements in algebraic number theory by proving the local Langlands conjecture for GLn over a p-adic field and exploring the decomposition group's influence on the l-adic cohomology of simple Shimura varieties. These interconnected problems build on Langlands's 1969 Washington lecture, offering a non-abelian generalization of local class field theory. The findings mark a crucial step forward in understanding these complex mathematical concepts.