Quadratic points on the Fermat quartic over number fields

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

💡 Research Summary

The paper studies quadratic points on the Fermat quartic curve F₄: X⁴+Y⁴=Z⁴ over arbitrary number fields K. A point P∈F₄(ℚ̄) is called K‑quadratic if the field generated by its coordinates has degree 2 over K. The authors focus on number fields K for which the two elliptic curves

 E₁: y² = x³ + 4x and E₂: y² = x³ – 4x

have Mordell–Weil rank 0 over K. Under this hypothesis they prove three main results:

1. Finiteness and computability – The set Γ₂(F₄,K) of K‑quadratic points is finite and can be explicitly computed. This follows from the fact that the Jacobian of the modular curve X₀(64) (which is ℚ‑isomorphic to F₄) decomposes as J₀(64) ≅ E₁ × E₂. Hence rank J₀(64)(K) = rank E₁(K) + rank E₂(K) = 0, so J₀(64)(K) is a finite group and the standard symmetric‑square embedding C(2)(K) → J(K) yields a finite search space.

2. Odd‑degree fields – If