The Maximal Rank of Elliptic Delsarte Surfaces

TL;DR

This paper identifies all elliptic curves over k(t) suitable for Shioda's method to compute their Mordell-Weil rank, expanding understanding of elliptic Delsarte surfaces and their ranks.

Contribution

It characterizes all elliptic curves over k(t) where Shioda's method applies and computes their Mordell-Weil ranks, providing a comprehensive classification.

Findings

All elliptic curves over k(t) applicable to Shioda's method identified.

Mordell-Weil ranks computed for these curves.

Enhanced understanding of the structure of elliptic Delsarte surfaces.

Abstract

Shioda described in his article from 1986 a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over k(t). In this article we find all elliptic curves over k(t) for which his method is applicable. For each of these curves we also compute the Mordell-Weil rank.