Tate–Shafarevich results for quartic twists in characteristic 2

The aim of this paper is to present families of elliptic surfaces defined over function fields of even characteristic having arbitrarily large Mordell–Weil rank. More precisely, we study the elliptic surface arising from the quartic twist of a supersingular elliptic curve defined over the field with two elements, using the function field of a maximal curve C admitting an order 4 automorphism. For the resulting elliptic surface, we provide a rank formula for its Mordell–Weil group in terms of the genera of C and another curve covered by C.