The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields

Abstract

It is shown that the quartic Fermat equation x4+ y4= 1 has nontrivial integral solutions in the Hilbert class field Σ of any quadratic field [Formula: see text] whose discriminant satisfies -d ≡ 1 (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in [Formula: see text], for p (> 7) a prime congruent to 7 (mod 8), but does have a nontrivial solution in the odd degree extension Σ of K. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in detail and the results are applied to prove several properties of the Weber singular moduli introduced by Yui and Zagier.