Rank stability of elliptic curves over ${\mathbb{F}}_q(t)$ in residue classes

Abstract

Fix a prime $\ell$. Let $K={\mathbb{F}}_q(t)$ be a global function field such that $\gcd (q,6)=1$ and $q\equiv 1(\mathrm{mod}\ell )$. Let $E$ be a non-isotrivial elliptic curve over $K$. Given a fixed monic polynomial $Q$ over ${\mathbb{F}}_q$, and assuming some mild conditions on $E$, we show that the rank of $E$ does not change with respect to a positive proportion of $\mathbb{Z}/\ell \mathbb{Z}$ extensions $K(\sqrt[\ell ]{f})/K$, as $f$ varies over the set of monic polynomials over ${\mathbb{F}}_q$ such that $f\equiv A(\mathrm{mod}Q)$ for any given polynomial $A$. We obtain this result by combining analytic and probabilistic techniques to study the distribution of certain prime Selmer groups of auxiliary abelian varieties constructed from these polynomials $f$.

arXiv comment26 Pages. An updated version containing additional results will be uploaded in the near future. Comments Welcome!