Mathematical Sciences: Dual Pair Correspondences Automorphic Forms and Hecke Algebras

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Abstract Savin 9623533 Savin will continue his work on exceptional dual pair correspondences, over p-adic and real fields, with applications to automorphic forms, in particular to the construction of a motive with Galois group G2. A purpose of this research is to get a better understanding of zeroes of polynomials. Almost everybody knows how to solve a quadratic equation. It is also possible, although it is less known, to find zeroes of any polynomial of degree 3 or 4. A higher degree polynomial, in general, can not be solved, but can be studied by attaching to it a mathematical object called the Galois group. Conversly, one can ask the following question: given a Galois group, can you find a polynomial corresponding to that group? The aim of this research is to give a positive answer to this question for a class of groups of type G2(p). The degrees of corresponding polynomials are, roughly speaking, the 6th power of the prime number p. For p=5 it is around 6 billion! This research will provide a good understanding of such polynomials.