Squares in arithmetic progression over quadratic extensions of number fields

We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we determine the set of $K$-quadratic points on this curve under certain conditions on the base field $K$. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of $K$ must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions.

💡 Research Summary

The paper investigates arithmetic progressions of squares in quadratic extensions of arbitrary number fields K. By translating the condition “a₁²,…,aₙ² form an arithmetic progression” into a system of quadratic equations, the author identifies a genus‑5 curve C whose K‑quadratic points correspond precisely to such progressions of length five. A parametrisation of C via a variable t leads to a pair of genus‑1 curves C₀: y²=G(t) with G(t)=t⁴−2t³+2t²+2t+1, and two natural maps φ₀, φ₀′ sending a point (t,X₁,X₃)∈C(L) to (t,X₁) and (−t,X₃) on C₀.

When t∉{0,±1}, the author introduces auxiliary quantities s=(t−1)/t and r=s², which satisfy two quadratic relations. These relations give rise to six genus‑1 curves C₁,…,C₆. Each Cᵢ is Q‑isomorphic to a specific elliptic curve (E₀, E₁, E_{±1,1}, E₄, E_{±1,6}, E₆) after an appropriate quadratic twist. Consequently, the set of K‑quadratic points on C can be studied through the rational points on these elliptic curves.

Two hypotheses on the base field K are imposed:

• cond A(K): the K‑rational points of the three elliptic curves E_{±1,1}, E₄, E_{±1,6} coincide with their ℚ‑rational points (including all quadratic twists).
• cond B(K): rankℤ E_{±1,1}(K)=0 and either