This is a beautiful paper, significant both mathematically (in terms of proof, technique, and structure revealed) and scientifically (in terms of questions raised and rigorously tested). It is a pioneering study on “critical” cubic equations; see [Y. Harpaz, Ann. Inst. Fourier 67, No. 5, 2167–2200 (2017; Zbl 1401.14125)] for a nice reference on the broader setting of log Calabi-Yau surfaces, in which the present paper lives.
Let \(M(x,y,z) = x^2+y^2+z^2-xyz\) be the Markoff polynomial. The authors show that for integers \(k\), the Markoff-type cubic surface \(M = k\) almost always satisfies the integral Hasse principle, but also sometimes fails it (so that “almost always” cannot be improved to “always”). As in past works of the same flavor, the proof is based on a variance-type analysis. But for “critical” equations such as \(M = k\), one cannot directly adapt older “subcritical” variance-type frameworks (such as those of R. C. Vaughan [Proc. Lond. Math. Soc. (3) 41, 516–532 (1980; Zbl 0446.10042)], J. Brüdern [Math. Scand. 68, No. 1, 27–45 (1991; Zbl 0759.11031)], and C. Hooley [Acta Arith. 173, No. 1, 19–39 (2016; Zbl 1359.11081)]). New issues, related to the square-root barrier in the classical circle method, emerge, in the most difficult piece (44) of the authors’ variance-type analysis.
At least generically, the authors handle the square-root aspect using the partially quadratic nature of the Markoff cubic, together with a mildly uniform version of the Kloosterman circle method for quadratic forms in \(4\) variables (for example, the delta method of [W. Duke et al., Invent. Math. 112, No. 1, 1–8 (1993; Zbl 0765.11038)] and [D. R. Heath-Brown, J. Reine Angew. Math. 481, 149–206 (1996; Zbl 0857.11049)] would suffice, up to uniformity questions that have been addressed in [N. Niedermowwe, J. Math. Sci., New York 171, No. 6, 753–764 (2010; Zbl 1282.11140)]). But in (44), there are also further multiplicative subtleties involving binary quadratic forms (see Lemma 9.5), which the authors address using [V. Blomer and A. Granville, Duke Math. J. 135, No. 2, 261–302 (2006; Zbl 1135.11020)].
One also has to produce, for typical integers \(k\), a lower bound on a suitable approximate singular series (36). The quantity (36) is related to the value of an \(L\)-function at \(s=1\) (see (61)), essentially since \(M=k\) is a surface. So (36) is more delicate to handle than similar quantities were in previous works.
It is worth mentioning several other aspects of the paper. The equations \(M=k\) define certain relative character varieties, and carry a rich group of nonlinear symmetries (Markoff morphisms, generated by permutations and Vieta moves), rich enough so that the integral points on \(M=k\) (for any \(k\)) lie in at most finitely many orbits. Thus the authors produce (for almost all \(k\)) not only integral points, but in fact a Zariski dense set of integral points (see §5). Furthermore, Theorem 1.1 provides an “explicit reduction (descent)” algorithm on \(M=k\), and this algorithm allows for rigorous numerical study of Hasse failures for \(k\) up to any finite threshold. This leads to the fascinating Conjecture 10.2(1) that the number of Hasse failures for \(0\le k\le K\) as \(K\to \infty\) is asymptotic to a constant times \(K^\theta\), for some \(\theta\in (\frac12, 1)\); the fact that \(\theta\) could be strictly greater than \(\frac12\) is striking, given that so far one only knows how to rigorously produce around \(O(K^{1/2})\) Hasse failures as \(K\to \infty\) (see e.g. the works [J.-L. Colliot-Thélène et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 21, 1257–1313 (2020; Zbl 1478.11091) and [D. Loughran and V. Mitankin, Int. Math. Res. Not. 2021, No. 18, 14086–14122 (2021; Zbl 1485.11106)] inspired by the paper under review). Note that there would be no convincing basis for this conjecture were it not for Theorem 1.1.