NSF Award
2401547
Number theory has a rich history of long-standing questions that are surprisingly easy to state but notoriously difficult to answer -- for example which sums of perfect powers equal another perfect power (a generalization of the famous Fermat's Last Theorem that remains unanswered). The central paradigm in arithmetic geometry is that the geometry of polynomial equations has a strong bearing on the geography of whole number solutions. In the last 20 years, the quadratic Chabauty method has emerged as a powerful new technique for locating whole number solutions (i.e. rational points) on curves that were impervious to all previous methods. In practice, one makes several simplifying assumptions on the curves in question to use this method. The PI will continue work with collaborators in the area of arithmetic geometry: exploring a new theoretical framework for the quadratic Chabauty method; explicitly computing invariants measuring the complexity of reduction types of curves; and introducing new computational tools to study the Ceresa cycle, a fundamental invariant associated to an algebraic curve with close ties to its geometry and arithmetic. Additionally, the PI will organize events intended to support and showcase the work of junior mathematicians at the institutional, regional, and national levels. <br/><br/>The proposed research will explore three aspects of the arithmetic and geometry of curves. One goal is to explicitly describe good models for solvable covers of the projective line over p-adic fields, and use them to extract various arithmetic invariants of these curves, building on past work by the PI for cyclic covers. Another goal is to build new algorithms for computing various constants appearing in quadratic Chabauty method, using a new framework at bad primes jointly developed with her collaborators, utilizing recent advances in the comparison of p-adic integration theories for curves with bad reduction. The third goal is to use techniques from p-adic integration and the geometry of curves in characteristic p to provide new methods for establishing the nontriviality of the Ceresa cycle, a canonical one dimensional algebraic cycle associated to a curve.<br/><br/>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.
