# Fermat's cubic equation in quadratic extension of $\mathbb{Q}$

Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $$\mathbb{Q}$$ based on the existence or not existence of non-trivial solutions of Fermat's cubic equation in the ring of integers belonging to those extensions? Considering d as the only number necessary to determine if $$\mathbb{Q}[\sqrt{d}]$$ has or not non-trivial solutions.

• See section 2 of this paper. – Wojowu Feb 26 at 20:38
• I've already checked that paper. The result described, in general case, is not unknown for me. However I was not able to describe my idea clearly: i) Non-trivial solutions are searched in $Z[\omega _d]$, the ring of integers of $Q(\sqrt{d})$ – E. Pech Feb 26 at 23:41
• ii) Criteria should be as simple as a modular congruence over Z. For example, "if $d\equiv 3(4)$ then there exist non-trivial solutions" Maybe I just can not see it, but I don't think Section 2 accomplish any of these points. – E. Pech Feb 26 at 23:49
• Than you for all your help. – E. Pech Feb 26 at 23:50

I don't want to toot my own horn, but I coauthored a paper on this topic with Marvin Jones. One direction of our result is conditional on the Birch and Swinnerton-Dyer conjecture (see the first remark on page 3). Basically, the Fermat cubic $$x^3 + y^3 = z^3$$ is isomorphic to $$E : y^2 + y = x^3 - 7$$. Since this curve has rank zero over $$\mathbb{Q}$$, the existence or non-existence of solutions in $$\mathbb{Q}(\sqrt{d})$$ is equivalent to the quadratic twist $$E_{d}$$ having positive rank or not over $$\mathbb{Q}$$. One can then give a criterion for this by relating $$L(E_{d},1)$$ to the Fourier coefficients of weight $$3/2$$ modular forms. The method is very similar to Tunnell's solution of the congruent number problem.