The research on this project will be carried out by two investigators, the principal investigator Herve Jacquet and his postdoctoral associate Michael Heumos. Jacquet will continue his efforts at generalizing his new trace formula. Results on special values of L-functions, the determination of the residual spectrum of the general linear group and a better understanding of the Weil representation are among the long range goals of the project. Heumos will study models for unitary representations of GLn over a p-adic field. These models generalize the standard Whittaker model and are subgroups of GLn. Examples suggest that perhaps all unitary representations embed uniquely in a unique model. This would provide a new perspective to the study of unitary representations of the p-adic GLn and would have widespread applications. This research is in the area of number theory commonly known as the "Langland's program". It applies advanced notions from many areas, most especially modern analysis, to answer fundamental questions in arithmetic.