On the Arithmetic of x3+y3=nz3 : Complex Multiplication and Asymptotic Distributions

Abstract

This expository article reviews the arithmetic of the Fermat elliptic curve , which admits complex multiplication by the Eisenstein integers . We outline the computation of Frobenius traces over finite fields via cubic residue characters and Jacobi sums, explicitly linking local Diophantine data to the associated Grössencharakter over . Following Deuring's theorem, we detail the identification of the Hasse-Weil L-function with a Hecke L-function and review the standard formulation of the Birch and Swinnerton-Dyer conjecture. Finally, we present a linear-time lattice algorithm for computing the sequence of local traces and derive an asymptotic expansion for the mean number of affine solutions.