Diophantine geometry and Galois representationsWangErickson, Carl (2020) Diophantine geometry and Galois representations. In: Pitt Momentum Fund 2020, University of Pittsburgh, Pittsburgh, Pa. (Unpublished)
Abstract"Finding integer solutions to polynomial equations, also known as “Diophantine geometry,” is a fundamental topic throughout the history of the study of numbers. In the early 20th century, one of Hilbert’s famous list of mathematical problems was to find an algorithm that would calculate the solutions to any polynomial equation. Around 1970, a group of mathematicians determined that there can be no such algorithm. Since we lack a general algorithm, it is natural to start with some particular kind of equations. Algebraic curves are a natural candidate: they are distinguished by the complex number solutions to the equation forming a surface (that is, a 2dimensional space). Remarkably, in the 1980s Faltings proved that when the number of “holes” in this surface is at least 2, then there are finitely many solutions. However, Faltings’s method gave no way to actually find the solutions. In particular, if you find some collection of solutions, how can you know that you have found them all? Starting around 2010, Minhyong Kim introduced a suite of new ideas to this endeavor, now known as the nonabelian Chabauty method. The goal of this project is to produce a framework that makes the nonabelian Chabauty method amendable to computation in more cases, thereby making it possible to find all of the solutions on more algebraic curves. " Share
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