Mathematical Sciences: Eisenstein Series, Theta Functions and Special Values of L-Functions

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This award supports the research in automorphic forms of Professor Steven Kudla of the University of Maryland at College Park. Dr. Kudla's project is to work on two problems involving theta functions, L-functions, and arithmetic. The first of these problems concerns a conjecture about the identification of the first two terms in the Laurent expansion of the Siegel Eisenstein series at certain special points. The second is concerned with an analogue for the triple L-function of the result of Gross and Zagier that relates the derivative of the standard L-function of a holomorphic cusp form on GL(2) to the height pairing of Heegner points on the Jacobian of the modular curve. 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.