The Algebra and Arithmetic of Splitting Fields

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This project will explore algebraic and geometric objects that have played an important role in the development of many areas of mathematics and physics, known as central simple algebras and their splitting fields. The theory of quaternion algebras, examples of these structures, was utilized throughout the nineteenth and twentieth centuries in classical mechanics, quantum mechanics, and the theory of relativity, with current applications in fields such as aerospace modeling, computer graphics, robotics, and wireless communication. In mathematics, central simple algebras and the associated Brauer group have provided a bridge for techniques to be exchanged between diverse areas of research, from number theory and arithmetic geometry, to topology and algebraic geometry. This project aims to broaden a relatively new frontier between the splitting fields of central simple algebras and the arithmetic of elliptic curves, which are themselves important mathematical objects. The project will also support new pedagogical tools, opportunities for public outreach and engagement, early career mentoring and training opportunities for undergraduate and graduate students, and cross-disciplinary collaboration.More precisely, this project centers around three main problems: the period-index problem for the Brauer group, the problem of splitting central simple algebras by genus one curves, and explicit presentations of moduli spaces of elliptic curves. The techniques utilized---toroidal geometry, deformation theory, and Hilbert schemes---are widely employed in algebraic geometry, but less systematically so in pure algebra. While much recent progress in the period-index problem for the Brauer group has been through leveraging tools from algebraic geometry to control ramification splitting, this project adopts novel tools such as reciprocity sequences to achieve improved results as well as indicate long-term conjectural targets. The problem of whether any Brauer class is split by the function field of a genus one curve, while having only gained attention in the past decade, turns out to have surprising and deep connections to other important problems in algebra, number theory, and algebraic geometry. The project initiates a wide-ranging program exploiting connections between this problem, the classical open problem of cyclicity of central simple algebras in prime degree, the period-index problem for abelian variety torsors, the arithmetic of modular curves, and the problem of explicit constructions of moduli spaces of elliptic curves.This project is jointly funded by the Algebra and Number Theory Program 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.