Number Theory Seminar
- Event Type
- Seminar/Symposium
- Sponsor
- Department of Mathematics
- Location
- Gregory Hall 205
- Date
- Nov 19, 2024 11:00 - 11:50 am
- Speaker
- Carlo Pagano (Concordia University)
- Contact
- Jesse Thorner and Yuan Liu
- jat9@illinois.edu and yyyliu@illinois.edu
- Views
- 69
- Originating Calendar
- Number Theory Seminar Calendar
- Title: Hilbert 10 via additive combinatorics
Abstract: 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 extension 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 use Green--Tao to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring.