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

TL;DR

This paper demonstrates that the quartic Fermat equation has nontrivial solutions in Hilbert class fields of certain imaginary quadratic fields, linking solutions to elliptic curve points and Weber singular moduli.

Contribution

It establishes the existence of solutions in Hilbert class fields for specific quadratic fields and connects these solutions to elliptic curve points and Weber singular moduli.

Findings

Solutions exist in Hilbert class fields for certain imaginary quadratic fields.

No solutions in the base quadratic fields for primes p ≡ 7 mod 8.

Explicit formulas for elliptic curve points of order 4 are derived.

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.