Arithmetic statistics and Diophantine stability for elliptic curves (Q6073935)

Let \(p\) be a prime number. For an elliptic curve \(E\) defined over \(\mathbb{Q}\) and a cyclic \(p\)-extension \(L\) of degree \(p^n\) over \(\mathbb{Q}\), the author defines the strong diophantine stability of \(E\) in \(L\) by \(E(\mathbb{Q}) = E(L_{\infty})\), where \(L_{\infty}\) is the cyclotomic \(\mathbb{Z}_p\)-extension of \(L\). Also, \(E\) is called strongly diophantine stable at \(p\) if, for any choice of \(n \ge 1 \) and a finite set \(\Sigma\) of primes, \(E\) is strongly diophantine stable in infinitely many \(\mathbb{Z}/p^n \mathbb{Z}\)-extensions \(L/\mathbb{Q}\) in which the primes in \(\Sigma \) splits. (The general notion of diophantine stability is introduced by \textit{B. Mazur} and \textit{K. Rubin} [Am. J. Math. 140, No. 3, 571--616 (2018; Zbl 1491.14036)]). The author similarly defines the \(\Sha \)-stability using the Tate-Shafarevich group instead of the Mordell-Weil group. The main result of this paper (Theorem 6.5) provides a conditional answer to the question of the portion of elliptic curves defined over \(\mathbb{Q}\) having strong diophantine stability and \(\Sha \)-stability. To be more precise, let \(\mathscr{E}_p\) be the set of isomorphism classes of elliptic curves \(E/\mathbb{Q}\) with height less than \(x\) that satisfy the following conditions: (i) \(\mathrm{rank} E(\mathbb{Q})=0\); (ii) \(E[p]\) is irreducible as a Galois module; (iii) \(E\) has good reduction at \(2\) and \(3\); (iv) \(E\) has good ordinary reduction at \(p\); (v) \(E\) is strongly diophantine stable and \(\Sha \)-stable. Then, the estimate of \[ \lim\inf_{x \to \infty} \dfrac{|\mathscr{E}_p (x)|}{|\{ E/\mathbb{Q} \text{ with height less than } x \}|} \] from below is explicitly given under certain conjectures. In addition to the stability results, the author also proves theorems on the growth of the rank of \(E\) and the order of \(\Sha\).