Cohomology of locally symmetric spaces and applications to number theory

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The cohomology of locally symmetric varieties plays an important role innumber theory and in particular Langlands's program. One reason is thatthey are examples of spaces exhibiting an action of both a Hecke algebra]and a Galois group. A fundamental goal is to relate these actions and thecorresponding L-functions. In previous work, the principal investigatorcreated a new tool, L-modules, to study such cohomology. In the currentproject, the principal investigator proposes to incorporate Hecke andGalois actions into L-modules. He will also prove a "decompositiontheorem" for L-modules and apply it to construct cycles associated tosub-locally symmetric varieties.The proposed research studies symmetry in geometry and number theory.Applications of geometry and number theory abound, for example tocryptography and crystallography. A symmetry in geometry is atransformation of space which doesn't change distance, for example arotation. In number theory a symmetry is a transformation of a numbersystem that transforms the sum of two numbers into the sum of thetransforms and likewise for the product, for example complex conjugation.In both subjects, the most interesting objects are those which remainunchanged under many symmetries. For example, in geometry the sphere isinvariant under all rotations, while in number theory the set of roots of apolynomial equation with integer coefficients are invariant under allsymmetries. The investigator's research concern higher-dimensional objectswhich possess both geometric and number theoretic symmetries, and therelations between these symmetries.