This award supports the research in Automorphic Forms by Professor Jonathan Rogawski of the University of California at Los Angeles. Dr. Rogawski's project is to continue his investigation of the relations between Number Theory and the theory of automorphic forms on unitary groups in three variables. He plans three interrelated lines of research: The first involves a continuation of his collaboration with S. Gelbart on the representation of the L-function by an integral of Rankin-Shimura type. The second concerns p-adic families of automorphic representations on U(3) and their relation with Galois representations. The third is the investigation of period relations, special values of L-functions, and the Beilinson conjectures for Shimura varieties associated to U(3). Non-Euclidean plane geometry began in the early nineteenth century as a mathematical curiosity, but by the end of that century, mathematicians had realized that many objects of fundamental importance are non-Euclidean in their basic nature. The detailed study of non-Euclidean plane geometries has given rise to several branches of modern mathematics, of which the study of Modular and Automorphic Forms is one of the most active. This field is principally concerned with questions about the whole numbers, but in its use of Geometry and Analysis, it retains connection to its historical roots.
