Baker-Gross theorem revisited

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.