Residual Representations, Relative Trace Formulas, Fourier Coefficients of Eisenstein Series

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ABSTRACT Roger Howe, Dihau Jiang Yale University 98 03617 This project tackles several problems in an area of number theory known as the Langlands Program. One problem concerns the periods of automorphic forms and special values of L-functions. The investigators also have some applications of residual representations and trace formula. The third part of the study concerns Fourier coefficients of Eisenstein series and prehomogeneous vector spaces. One of the most important mathematical ideas of the second half of the century, is that analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, you might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kind of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kind of analytic function can contain the answer or answers to the discrete examples about. Actually the first examples of this were discovered a couple hundred years ago, but in the past thirty years, these examples have been systematized into a branch of number theory called the Langlands program.