Parity of ranks of elliptic and hyperelliptic curves

Research Area: Number Theory

The study of rational solutions to polynomial equations is a subject that is at the heart of algebraic number theory. Abelian varieties (AVs) are those equations for which the solutions form an abelian group: there is a method for creating new solutions from known ones. The classical and simplest case is that of elliptic curves, such as y^2+y=x^3-x, where the trivial solution x=y=0 lets one generate an infinite number of new solutions. The central questions about elliptic curves and AVs remain unresolved, although the Birch-Swinnerton-Dyer (BSD) conjecture provides an unproven recipe for extracting the critical arithmetic data (the "rank"). This conjecture is regarded as one of the central unsolved problems in modern mathematics, and is one of the seven Clay Millenium Problems (now six, after the Poincare conjecture was solved by Perelman).

The project is a new investigation into the parity of ranks of elliptic curves and higher dimensional AVs - Jacobians of hyperelliptic curves. The theory of elliptic curves is relatively well-developed and can be used as a testing ground when studying higher dimensional AVs. Hyperelliptic curves have seen a number of breakthroughs in the last few years, which will provide very useful tools for this project: these include explicit methods for working with root numbers (M. Bisatt), Tamagawa numbers (A. Betts), differentials (S. Kunzweiler) and local Galois representations (T. Dokchitser, V. Dokchitser, C. Maistret, A. Morgan). Furthermore, both elliptic and hyperelliptic curves share the feature that they are suitable for doing numerical experiments, which can often both support and help to guide the theoretical work.

The general aim of the project is to develop new techniques for working with higher genus curves and their invariants related to the BSD conjecture in the context of explicit number theory. Specific goals are to: (i) investigate the distribution of the parity of ranks for families of curves, and (ii) develop methods that exploit maps between curves of different genus to control important arithmetic data, such as ranks and Selmer groups.