Rational points on hyperelliptic curves via nonabelian descent

Abstract: 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 $f$.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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