2018/01/18 by Kirti Joshi, Joshi, Kirti
Computer Science · Mathematics · #11G05 #14G05 #14H52 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1801.06245
openalex publication_date 2018/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
I provide a systematic construction of points, defined over finite radical\nextensions, on any Legendre curve over any field of characteristic not equal\ntwo. This includes as special case Douglas Ulmer's construction of rational\npoints over a rational function field in characteristic p>0. In particular I\nshow that if n\≥ 4 is any even integer and not divisible by the\ncharacteristic of the field then any elliptic curve E over this field has at\nleast 2n rational points over a finite solvable field extension. Under\nadditional hypothesis, when the ground field is a number field, I show that\nthese are of infinite order. I also show that Ulmer's points lift to\ncharacteristic zero and in particular to the canonical lifting.\n