SFU Number Theory and Algebraic Geometry Seminar: Netan Dogra

Let $f(x)$ be a separable polynomial with rational number coefficients. In this talk I will review how the rational points of the hyperelliptic curve $y^2 = f(x)$ can sometimes be determined using the number field obtained by adjoining a root of $f$, via the Chabauty--Coleman method and the theory of the $2$-Selmer group. I will then explain the limitations of this method, and how to give a `nonabelian' generalisation. The punchline will be that, if the Chabauty--Coleman method doesn't work, we can sometimes determine the rational points using the field obtained by adjoining two roots of $f$.