Arithmetic statistics and diophantine stability for elliptic curves

Abstract

We study the growth and stability of the Mordell–Weil group and Tate–Shafarevich group of an elliptic curve defined over the rationals, in various cyclic Galois extensions of prime power order. Mazur and Rubin introduced the notion of diophantine stability for the Mordell–Weil group an elliptic curve $$E_{/{{\mathbb {Q}}}}$$ at a given prime p. Inspired by their definition of stability for the Mordell–Weil group, we introduce an analogous notion of stability for the Tate–Shafarevich group, called -stability. From the perspective of Iwasawa theory, it benefits us to introduce a stronger notion of diophantine stability for the Mordell–Weil group. It is shown that any non-CM elliptic curve of rank 0 defined over the rationals is strongly diophantine stable and -stable at $$100\%$$ of primes p. Next, we show that standard conjectures on rank distribution give lower bounds for the proportion of rational elliptic curves E that are strongly diophantine stable at a fixed prime $$p\ge 11$$ . Related questions are studied for rank jumps and growth of ranks Tate–Shafarevich groups on average in prime power cyclic extensions.