By Michael Harris
This ebook goals first to turn out the neighborhood Langlands conjecture for GLn over a p-adic box and, moment, to spot the motion of the decomposition workforce at a chief of undesirable relief at the l-adic cohomology of the "simple" Shimura kinds. those difficulties cross hand in hand. the consequences symbolize a big increase in algebraic quantity thought, eventually proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of neighborhood classification box theory.
The neighborhood Langlands conjecture for GLn(K), the place okay is a p-adic box, asserts the life of a correspondence, with definite formal homes, touching on n-dimensional representations of the Galois team of K with the illustration thought of the in the neighborhood compact workforce GLn(K). This publication constructs a candidate for one of these neighborhood Langlands correspondence at the vanishing cycles connected to the undesirable relief over the integer ring of K of a undeniable relations of Shimura kinds. And it proves that this is often approximately suitable with the worldwide Galois correspondence discovered at the cohomology of an identical Shimura forms. The neighborhood Langlands conjecture is got as a corollary.
Certain thoughts constructed during this booklet may still expand to extra common Shimura types, delivering new situations of the neighborhood Langlands conjecture. furthermore, the geometry of the certain fibers is precisely analogous to that of Shimura curves and will be anticipated to have purposes to quite a few questions in quantity theory.
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