The challenge mentioned in the title is to determine the set of nonzero integer solutions to \(X^4+Y^4=17Z^4\), or equivalently to find the set of rational points on the genus-\(3\) projective curve \(\mathcal D\) defined by this equation. The authors prove that the eight obvious rational points with \(\{| X/Z| ,| Y/Z| \}=\{1,2\}\) are the only ones.
G. Faltings [Invent. Math. 73, 349–366 (1983; Zbl 0588.14026)] proved that any curve \(X\) of genus at least \(2\) over a number field \(k\) has finitely many \(k\)-rational points. But there is no algorithm currently known that is guaranteed to determine \(X(k)\) explicitly in all cases, even when the genus is as small as \(3\). Nevertheless there exist techniques, mostly predating Faltings’ work, that often succeed in determining \(X(k)\) for individual curves \(X\), on a case-by-case basis. These techniques have been made explicit by several authors in recent years.
Two such methods, based on ideas of Demyanenko and Chabauty respectively, do not apply directly to the curve \(\mathcal D\) at hand, because, as Serre knew, the Mordell-Weil ranks of the quotients of the Jacobian of \(\mathcal D\) are too large. The authors must therefore use a combination of methods.
First the problem of determining \({\mathcal D}({\mathbb Q})\) is reduced to the problem of determining the rational points on a finite set of degree-\(2\) unramified covers (of genus \(5\)). In fact, because of automorphisms of \(\mathcal D\), it suffices to examine one such cover. This genus-\(5\) curve covers a genus-\(2\) curve \(\mathcal C\), so it remains to determine \({\mathcal C}({\mathbb Q})\). Unfortunately the Mordell-Weil rank of the Jacobian of \(\mathcal C\) is too large for Chabauty’s method to apply directly to \(\mathcal C\). But \(\mathcal C\) over the field \(K={\mathbb Q}(\sqrt{2},\sqrt{34})\) has a degree-\(2\) unramified cover mapping back down to a genus-\(1\) curve \(\mathcal F\) over \(K\), and one shows that points of \({\mathcal C}({\mathbb Q})\) give rise to points in \({\mathcal F}(K)\) with \(x\)-coordinate in \(\mathbb Q\) (for a particular model of \(\mathcal F\)). The set of such points on \(\mathcal F\) are determined by “elliptic Chabauty”, a variant of Chabauty’s method, developed in [E. V. Flynn and J. L. Wetherell, Manuscr. Math. 100, 519–533 (1999; Zbl 1029.11024) and N. Bruin, Chabauty methods and covering techniques applied to generalised Fermat equations, CWI Tracts 133, Amsterdam (2002; Zbl 1043.11029)], independently.
As the authors suggest, the particular arguments used here probably will apply to some other curves of the form \(X^4+Y^4=cZ^4\) with \(c \in {\mathbb Q}^*\), although these must be considered one at a time: success depends on the Mordell-Weil ranks of the auxiliary curves, and these ranks are hard to predict in advance. More generally, the techniques of using unramified covers and Chabauty methods should apply to many other curves.