Tate-Shafarevich results for quartic twists in characteristic $2$

Abstract

The aim of this paper is to present elliptic curves defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study elliptic curves arising as quartic twist of a supersingular elliptic curve defined over $\mathbb{F}_2$ using the function field of a maximal curve $C$ that admits an order 4 automorphism. For such elliptic curves we provide a rank formula for its Mordell-Weil group in terms of the genera of $C$ and of another curve covered by $C$.