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Irrational points on random hyperelliptic curves

Banff International Research Station for Mathematical Innovation and Discovery

Let $d$ and $g$ be positive integers with $1< d < g$. If $d$ is odd, we prove there exists $B(d)>0$ such that a positive proportion of odd genus $g$ hyper elliptic curves over $\mathbf{Q}$ have at most $B(d)$ points of degree $d$. If $d$ is even, we similarly bound the degree $d$ points not pulled back from degree $d/2$ points of the projective line. Our proof proceeds by refining Parkâ s recent application of tropical geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves. This is joint work with Joseph Gunther.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: Emory University

2019-03-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68694

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/