Rank stability of elliptic curves over prime cyclic extensions of function fields

Abstract

Let $\ell$ be an odd prime number. Let $E$ be a non-isotrivial elliptic curve defined over the global function field $K = \mathbb{F}_q(t)$ such that $(q, 6) = 1$ and $q \equiv 1 \mod \ell$. We show that as $q$ grows arbitrarily large, both analytic and algebraic ranks of $E$ do not grow upon base change to $100\%$ of $\mathbb{Z}/\ell \mathbb{Z}$ Galois extensions over $K$. One of the applications we obtain is that over a positive proportion of $\mathbb{Z}/\ell \mathbb{Z}$ Galois extensions $L$ over $K$, the Tate--Shafarevich group of $E$ over $L$ is non-trivial. To obtain these results, we incorporate several geometric and probabilistic techniques which were studied in previous literature. These include the construction of desired auxiliary abelian varieties by Mazur--Rubin--Silverberg, the theory of middle convolution and big monodromy results by Katz and Hall, uniform Chebotarev density theorem by Kowalski, and distribution of prime Selmer groups of auxiliary abelian varieties by Klagsbrun--Mazur--Rubin and Park.