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

Abstract

It is shown that the quartic Fermat equation $x^4+y^4=1$ has nontrivial integral solutions in the Hilbert class field $\Sigma$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d\equiv 1$ (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in $K=\mathbb{Q}(\sqrt{-p})$, for $p$ $(>7)$ a prime congruent to $7$ (mod 8), but does have a nontrivial solution in the odd degree extension $\Sigma$ 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.

Cited by in corpus (3)

• Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)
• Product formulas for the $5$-division points on the Tate normal form and the Rogers-Ramanujan continued fraction
• Periodic points of algebraic functions and Deuring's class number formula