Baker-Gross theorem revisited

TL;DR

This paper provides an alternative proof of the Baker-Gross theorem concerning meromorphic solutions of the Fermat cubic, introduces new formulas, and discusses implications for higher-degree Fermat curves.

Contribution

It offers a novel proof of the Baker-Gross theorem and derives additional formulas, expanding understanding of solutions to Fermat cubic equations.

Findings

Alternative proof of Baker-Gross theorem

New explicit formulas for solutions

Remarks on Fermat curves of higher degree

Abstract

F. Gross conjectured that any meromorphic solution of the Fermat Cubic $F_3\colon\ x^3+y^3=1$ are elliptic functions composed with entire functions. The conjecture was solved affirmatively first by I. N. Baker who found explicit formulas of those elliptic functions and later F. Gross gave another proof proving that in fact one of them uniformize the Fermat cubic. In this paper we give an alternative proof of the Baker and Gross theorems. With our method we obtain other analogous formulas. Some remarks on Fermat curves of higher degree is given.