Modular Cocycles, Explicit Class Field Theory, and Quantum Designs

Recent research developments by the PI and others have revealed a connection between major open problems in number theory and quantum information theory. In number theory, Hilbert's 12th Problem (one of a famous list of 23 problems proposed in 1900) and the related Stark conjectures (formulated in the late 1970s) concern finding more explicit expressions for certain abstract structures involving algebraic numbers. In quantum information theory, Zauner's conjecture (1999) predicts the existence of highly regular geometric configurations called SICs, which describe quantum measurements. The research project involves investigating a connection between the Hilbert/Stark problems and Zauner's conjecture to provide new insights in both areas. Results from the project will allow for faster computation of SICs, which have potential applications to quantum state tomography as well as classical compressed sensing for radar. The project will also support a graduate student's involvement, allow the PI and his students to disseminate their work through conferences and seminars, support a research seminar in number theory at the PI's institution, and support outreach work by the PI to high school students. In the project, the PI will prove new results on complex analytic modular cocycles to refine the statement of the Stark conjectures for real quadratic fields and (through joint work with Appleby and Flammia) to conjecturally construct symmetric informationally complete positive operator-valued measures (SIC-POVMs or SICs) in every dimension. The project will develop the theory of (generalized, multiplicative) analytic modular cocycles and their "real multiplication values," reformulating (and proving results towards) the Stark conjectures in a language similar to the language of modular forms rather than L-functions. It will produce a geometric interpretation of explicit class field theory for real quadratic fields through structures generalizing SICs. Connections will be explored to Eisenstein and Shintani cocycles as studied by Charollois, Dasgupta, Greenberg, Hill, Sczech, and Solomon and to p-adic rigid meromorphic cocycles as studied by Darmon, Pozzi, and Vonk. Connections to quantum field theory will also be explored, and generalized beta integral relations from the mathematical physics literature will be applied to modular cocycles and their real multiplication values. This project is jointly funded by the Algebra and Number Theory in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.