A method for construction of rational points over elliptic curves II: Points over solvable extensions

Abstract

I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over a rational function field in characteristic $p>0$. In particular I show that if $n\geq 4$ is any even integer and not divisible by the characteristic of the field then any elliptic curve $E$ over this field has at least $2n$ rational points over a finite solvable field extension. Under additional hypothesis, when the ground field is a number field, I show that these are of infinite order. I also show that Ulmer's points lift to characteristic zero and in particular to the canonical lifting.