Number Theory and Related Fields

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ABSTRACT for award DMS-0403374Barry Mazur The investigator is working on four projects: To understand the solutions of cubic equations in the plane (i.e., points on elliptic curves) over number fields and how the collection of these solutions grows as one enlarges the number field. To study, by algebraic geometric methods as well as (p-adic and classical) analytic methods the nature of congruences between coefficients of Fourier coefficients of modular forms, a subject pioneered by Euler, Jacobi, Ramanujan and others. To understand the implications to arithmetic of the new work on rationally connected algebraic varieties. To make use of the connection between differential topology, homotopy theory and the arithmetic theory of modular forms (the work of Michael Hopkins) to explore issues in classical homotopy theory. ( Michael Hopkins and the PI are currently trying to write up a proof of the fact that the spectrum PL/O that classifies differential structures on compact combinatorial manifolds and the "kernel of the Eisenstein ideal" in the spectrum known as tmf ("topological modular forms") at level 2 are canonically isomorphic after appropriate completion at an odd prime number p; this would allow one to apply deep arithmetic results to analyze the structure of these spectra).The fields of arithmetic, algebraic geometry, complex and p-adic analysis, differential topology, and homotopy theory have-currently-exciting points of contact, where new ideas in one realm allow one to make inroads in others. All four projects in which the PI is engaged have the aim of strengthening these connections. To take one example, the most classical issue of finding rational solutions of cubic equations in two variables- which now is the mainstay of the theory and practice behind public-key encryption- finds itself as the core problem, center stage in all four of the PI's projects, even though these projects range through all the fields we have listed above. To take another example, the classical problem pioneered by Euler, Jacobi, Ramanujan and others, of finding and explaining congruences between Fourier coefficients of modular forms- one of the PI's projects- is now most powerfully addressed by viewing the modular forms in question as a dense set of points in a beautiful (p-analytic) space, whose geometric features are as intriguing as they are vital for an understanding of these congruences, as well as for other equally basic questions in analytic number theory.