IRFP: Mass equidistribution on compact arithmetic surfaces and applications

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The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four month research fellowship by Dr. Paul Nelson to work with Dr. Philippe Michel at Ecole Polytechnique Federale de Lausanne in Lausanne, Switzerland.The project concerns variants of the arithmetic quantum unique ergodicity conjecture, which conjecturally describes the interplay between the quantum and classical dynamics of certain chaotic systems that arise from number-theoretic considera¬tions; many of the problems in this field follow from suitable generalizations of the Riemann hypothesis (a central unresolved problem in mathematics) but are considerably more acces¬sible, and so provide fertile grounds for the development of new techniques. The project consists of several closely-related components in which the PI will continue the line of re¬search initiated in his thesis, become exposed to new methods introduced by the host and his coauthors, and collaborate with the host on a number of projects of common interest.The PI will study the mass distribution of holomorphic eigenforms on compact arithmetic surfaces, for which his thesis work has substantially prepared him; the PI will learn in greater depth some of the techniques recently pioneered by the host and his collaborators, which should help him in this pursuit. Progress on this problem will likely require a new method for studying modular forms on compact arithmetic surfaces that substitutes for the use of Fourier expansions at the cusps of non-compact arithmetic surfaces; the PI and host would want to investigate further such a method in other contexts.The PI and the host will study in depth the arithmetic geometric applications of the methods and results developed in the thesis of the PI concerning the mass equidistribution of cusp forms of large level, with a view towards understanding the zero divisors of the modular parameterizations of strong Weil curves of large conductor as well as congruences between modular forms; this will involve techniques in which the host is an expert, build upon recent work of the PI, and contribute significant understanding of the arithmetic and geometric consequences of the analytic theory of modular forms.The PI will teach and mentor students at EPFL, thereby broadening the group of students with which he has interacted beyond those at Caltech who he has already had the privilege to instruct in several capacities: as a teaching assistant, as an instructor for courses that he has designed from the ground up, and as an official mentor on supervised independent research projects. He will enhance the general infrastructure for research by producing and making available written accounts of courses taught by him and others, as he has done for the past several years at Caltech. The project will result in lasting partnerships between the PI and leading researchers overseas, thereby strengthening the channels of communication. The results obtained will be disseminated broadly through online preprint servers and publication in mathematical research journals.