Arithmetic statistics and diophantine stability for elliptic curves

TL;DR

This paper investigates the stability and growth of the Mordell-Weil and Tate-Shafarevich groups of elliptic curves over rationals in cyclic Galois extensions, introducing new stability notions and providing average-case results using arithmetic statistics and Iwasawa theory.

Contribution

It introduces the concept of Sha-stability for Tate-Shafarevich groups and analyzes diophantine stability of elliptic curves on average, including fixed curve and conjectural rank distribution results.

Findings

Non-CM elliptic curves of rank 0 are diophantine and Sha-stable at 100% of primes p.

Under conjectures, a positive proportion of rational elliptic curves are stable at fixed primes p ≥ 11.

Results on rank jumps and Tate-Shafarevich group growth in prime power cyclic extensions.

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 $E$ 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 "Sha"-stability. Using methods in arithmetic statistics and Iwasawa theory, we study the diophantine stability of elliptic curves on average. First, we prove results for a fixed elliptic curve $E$ and varying prime $p$. It is shown that any non-CM elliptic curve of rank 0 defined over the rationals is diophantine stable and "Sha"-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 diophantine stable at a fixed prime $p\geq 11$. Related questions are studied for rank jumps and growth of ranks Tate-Shafarevich groups on average in prime power cyclic extensions.