Number/Representation Theory

Mazur--Rubin asked to what extent one can reconstruct a curve $C$ over a number field $K$, from the $\overline{K}$-points viewed as Galois set. More precisely they asked to what extent one can reconstruct $C$ over $\overline{K}$ from the set of number fields where $C$ acquires new points, and they provided some evidence of a positive answer in the case of curves of genus $0$. In this joint work with Zev Klagsbrun, we give an affirmative answer for a generic pair of elliptic curves having full rational $2$-torsion over $K$. The method employed is the combination of additive combinatorics and descent introduced by Koymans and the speaker in 2024, in their resolution of Hilbert 10th problem for finitely generated rings. I shall, along the way, overview the method and its several recent applications.