Curves with many degree d points

Joint with the MIT number theory seminar, note the special time and location

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.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar