Curves with many degree $d$ points

Abstract: When does a nice curve $X$ over a number field $k$ have infinitely many closed points of degree $d$? Faltings' theorem allows us to rephrase this problem in purely algebro-geometric terms, though the resulting geometric question is far from being fully solved. Previous work gave easy to state answers to the problem for degrees $2$ (Harris-Silverman) and $3$ (Abramovich-Harris), but also uncovered exotic constructions of such curves in all degrees $d \geqslant 4$ (Debarre-Fahlaoui). I will describe recent progress on the problem, which answers the question in the large genus case. Along the way we uncover systematic explanations for the Debarre-Fahlaoui counstructions and provide a complete geometric answer for $d \leqslant 5$. The talk is based on joint work with Isabel Vogt.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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