Derivatives of Eisenstein Series L - Functions and the Theta Correspondence

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This award supports an investigation into three problems involving the theta correspondence, the Siegel-Weil formula, values and derivatives of Eisenstein series and L-functions, and algebraic cycles on Shimura varieties. Problem I concerns the basic properties of the local theta correspondence, especially the question of the first occurence. Problem II concerns a possible relation between the height pairings of algebraic cycles on certain Shimura varieties and the Fourier expansions of central derivatives of Siegel Eisenstein series. Problem III involves the application of the relation of Problem II to the derivative, at the center of symmetry, of certain Langlands L-functions, generalizing the formula of Gross and Zagier. The research falls in the general mathematical area of the Langlands program.The Langlands program is part of Number Theory, which is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.