Quadratic points on the Fermat quartic over number fields
Enrique González-Jiménez
TL;DR
The paper develops a finite, computable framework for determining all $K$-quadratic points on the Fermat quartic $F_4: X^4+Y^4=Z^4$ over number fields $K$ under the condition that the elliptic curves $E_1: y^2=x^3+4x$ and $E_2: y^2=x^3-4x$ have rank $0$ over $K$. It combines a modular perspective (via the isomorphism $F_4 \simeq X_0(64)$ and $J_0(64)\simeq E_1^2\times E_2$) with a Mordell-inspired, field-by-field procedure that reduces the problem to the torsion structure of $E_1(K)$ and $E_2(K)$ and a descent on associated genus one curves. A key auxiliary result is that, for $j=1728$ curves, torsion cannot grow over odd-degree extensions, which allows reducing odd-degree cases to $\mathbb{Q}$; for degrees $<8$, the set $\Gamma_2(F_4,K)$ is determined by a finite collection of base fields $L$ and explicit primitive quadratic points $S^0_2(F_4,L)$. The contributions yield an explicit, implementable method to compute all $K$-quadratic points in these settings and connect the problem to a modular-elliptic framework with practical computational outcomes.
Abstract
Let $C$ be a curve defined over a number field $K$. A point $P\in C(\overline{\mathbb{Q}})$ is called $K$-quadratic if $[K(P):K]=2$. Let $K$ be a number field such that the rank of the elliptic curves $E_1:\,y^2= x^3 + 4x$ and $E_2:\,y^2= x^3 - 4x$ over $K$ are $0$. Under the above condition, we prove that the set of $K$-quadratic points on the Fermat quartic $F_4\colon X^4+Y^4=Z^4$ is finite and computable and we provide a procedure to compute this finite set. In particular, we explicitly compute all the $K$-quadratic points if $[K:\mathbb{Q}]<8$. Moreover, if the degree of $K$ is odd, we prove that all the $K$-quadratic points corresponds just to the $\mathbb{Q}$-quadratic points