4 papers
Hilbert's tenth problem for finitely generated rings
Peter Koymans, Carlo Pagano
This expository article covers the recent developments surrounding Hilbert's tenth problem for finitely generated rings. We start by recounting the history of Hilbert's tenth probl…
Hilbert's tenth problem via additive combinatorics
Peter Koymans, Carlo Pagano
For all infinite rings that are finitely generated over , we show that Hilbert's tenth problem has a negative answer. This is accomplished by constructing elliptic…
Elliptic curves of rank one over number fields
Peter Koymans, Carlo Pagano
We prove that for every number field , there exist infinitely many elliptic curves over with rank exactly equal to 1.
-torsion and integral points on quartic threefolds
Stephanie Chan, Peter Koymans, Carlo Pagano +1
We prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine qu…