Hilbert 10 via additive combinatoricsOberseminar Logik

by Carlo Pagano (MPIM/Concordia)

Monday May 19, 2025, 5:00 PM → 6:00 PM Europe/Berlin

Endenicher Allee 60, Nebengebäude/N0-003 - Seminarraum (Mathezentrum)

I will overview recent joint work with Peter Koymans, where we introduced a new method to construct elliptic curves over general number fields with positive but constrained rank. This comes in two flavors: either positive rank that is unchanged under quadratic extensions (which we established in 2024) or rank exactly equal to 1 (ongoing work in progress). Our main idea is to combine 2-descent with additive combinatorics. With the first of these two cases, we were able to settle Hilbert's tenth problem in the negative for all finitely generated infinite commutative rings. I will overview the reductions of Hilbert tenth's problem to such an elliptic curve question, and the main steps in our method.