A positive density of elliptic curves are diophantine stable in certain Galois extensions

Let $p \in {3, 5}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $ E $ defined over $ \mathbb{Q} $, ordered by height, that are diophantine stable in $ L $.

💡 Research Summary

The paper investigates the prevalence of elliptic curves over ℚ that remain “diophantine stable’’ in a fixed cyclic p‑extension L/ℚ, where p∈{3,5}. A curve E is diophantine stable in L if the natural inclusion induces an equality of rational points, i.e. E(L)=E(ℚ). Building on Mazur–Rubin’s notion of diophantine stability, the authors fix a Galois extension L/ℚ with Galois group Z/pZ and ask how many curves, ordered by height, satisfy the stability condition.

The core of the argument is a Selmer‑group analysis. Assuming that the p‑Selmer group Selₚ(E/ℚ) is trivial, one has rank E(ℚ)=0, no p‑torsion in E(ℚ), and trivial p‑primary Tate–Shafarevich group. The authors formulate a list of local conditions (Assumption 2.1) that guarantee, for each prime v in a finite set S (including all primes above p, all primes of bad reduction, and all primes ramified in L), the restriction map
γ_v : H¹(ℚ_v,E)