Rank growth of elliptic curves over $N$-th root extensions

Abstract

Fix an elliptic curve E over a number field F and an integer n which is a power of 3. We study the growth of the Mordell-Weil rank of E after base change to the fields d). If E admits a 3-isogeny, we show that the average "new rank" of E over K d , appropriately defined, is bounded as the height of d goes to infinity. When n = 3, we moreover show that for many elliptic curves E/Q, there are no new points on E over Q( 6 √ d), for a positive proportion of integers d. This is a horizontal analogue of a well-known result of Cornut and Vatsal. The proof combines our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the "correlation trick", which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces. As a corollary, we give bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.