Distribution of rational points and automorphic forms

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The PI proposes four research projects, the first three of which are related to the study of rational points of bounded height on certain classes of algebraic varieties in the context of Manin's conjecture for Fano varieties. In continuation of his joint work with Shalika and Tschinkel, the PI plans to study the distribution of rational points of bounded height on spherical varieties, and also compactifications of certain non-reductive algebraic groups. The fourth project, joint with a student, will be concerned with the distribution of orders in number fields. The proposed research, especially in the first three projects, is part of a broader program of bringing recent advances in the theory of automorphic forms to bear on the questions of arithmetic interest. The research theme applies methods from the theory of automorphic forms and ideas from arithmetic geometry to the study of rational points on homogeneous varieties. The PI believes that this research will advance knowledge and understanding of the arithmetic of higher dimensional varieties.Diophantine equations have been of interest since the antiquities. Often times a fundamental question of interest is whether a given Diophantine equation has solutions, or, if does, how many. In this research we propose to study certain classes of Diophantine equations with large groups of symmetries. The type of Diophantine equations we consider in this research a priori have an infinite number of solutions, so one desires a better understanding of the distribution of solutions. We propose to give approximate formulae for the number of solutions with bounded "height" - where here, "height" is a convenient measure of arithmetic complexity. We have also included an educational program in the proposal. Our previous work on the subject has produced a large number of concrete research problems accessible to undergraduate and graduate students. It has also led to the writing of a textbook joint with Steven J. Miller. We plan to develop these pedagogical programs further in the form of writing an advanced graduate textbook..