Hilbert 10 via additive combinatorics
SEMINARS, SEMINARS: NUMBER THEORY
When: March 26, 2025
3:00 pm - 4:00 pm
Where: Science Center 507
Address:
1 Oxford Street, Cambridge, MA 02138, United States
Speaker: Carlo Pagano (Concordia University)
In 1970 Matiyasevich, building on earlier work of Davis–Putnam–Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert’s 10th problem is undecidable for Z. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields L/K, an elliptic curve E/K with rk(E(L))=rk(E(K))>0.
In this talk I will explain joint work with Peter Koymans, where we combine Green–Tao with 2-descent to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring. The background material used to execute 2-descent in a quadratic twist will be explored during the pre-talk.
Pretalk title: Selmer groups and quadratic twisting
Pretalk abstract: The effect of quadratic twisting on Selmer groups of elliptic curves has been thoroughly explored in the last few decades, starting with work of Swinnerton-Dyer, and developed, in a number of directions, by several authors (Klagsbrun—Mazur–Rubin, Heath-Brown, Kane, Smith, among others). In this pre-talk we shall explain the basic algorithm that allows to compute 2-Selmer of such quadratic twists over a general number field. This will serve as one of the main technical pillar to execute the proof discussed in the talk.