Mathematical Sciences: Dual Pair Correspondences Automorphic Forms and Hecke Algebras

Savin将继续他在p进域和实域上的特殊对偶对应的研究，并将其应用于自同构形式，特别是Galois群G2动机的构造。本研究的一个目的是为了更好地理解多项式的零。几乎每个人都知道如何解二次方程。虽然不太为人所知，但也有可能找到任何3次或4次多项式的零。一般来说，高次多项式是无法求解的，但可以通过在其上附加一个称为伽罗瓦群的数学对象来研究。相反，我们可以问以下问题：给定一个伽罗瓦群，你能找到一个与该群对应的多项式吗？本研究的目的是对一类G2(p)型群给出这个问题的肯定答案。对应的多项式的次数大致是质数p的6次幂。对于p=5，它大约是60亿次！这项研究将提供一个很好的理解这种多项式。

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.