Descent on Elliptic Curves and Hilbert's Tenth Problem

TL;DR

This paper uses descent via isogeny on elliptic curves to construct specific subrings of rationals where Hilbert's Tenth Problem is undecidable, advancing previous methods involving elliptic divisibility sequences.

Contribution

It introduces a novel descent approach on elliptic curves to establish undecidability in new subrings of rational numbers, building on Poonen's elliptic divisibility sequence techniques.

Findings

Constructed two complementary subrings of rationals with undecidable Hilbert's Tenth Problem

Developed a descent method on elliptic curves for undecidability proofs

Extended previous results using elliptic divisibility sequences

Abstract

Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers.