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We introduce -adic exponential sums associated to a Laurent polynomial . They interpolate all classical -power order exponential sums associated to . We establish the Hodge bound for the Newton polygon of -functions of -adic exponential sums. This bound enables us to determine, for all , the Newton polygons of -functions of -power order exponential sums associated to an that is ordinary for . We also study deeper properties of -functions of -adic exponential sums. Along the way, we discuss new open problems about the -adic exponential sum itself.
We define and study the Burnside quotient Green ring of a Mackey functor, introduced in our 1990 MSRI preprint. Some refinements of Dress induction theory are presented, together with applications to computation results for -theory and -theory of finite and infinite groups.
Every finite-dimensional representation of an algebraic group gives a trace symmetric bilinear form on the Lie algebra of . We give criteria in terms of root system data for the existence of a representation such that this form is nonzero or nondegenerate. As a corollary, we show that a Lie algebra of type over a field of characteristic 5 does not have a “quotient trace form”, answering a question posed in the 1960s.
We compute dimensions and characters of the components of the operad of two compatible associative products and give an explicit combinatorial construction of the corresponding free algebras in terms of planar rooted trees.
Let be a smooth algebraic group acting on a variety . Let and be coherent sheaves on . We show that if all the higher sheaves of against -orbits vanish, then for generic , the sheaf vanishes for all . This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman–Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.