In 1983, Gerd Faltings proved a conjecture that had been open since 1922: any algebraic curve of genus two or higher, defined over a number field, has only finitely many rational points. The result earned Faltings a Fields Medal and settled a question that had organized much of twentieth-century number theory. It also left a gap that would persist for forty-two years. Faltings proved the number is finite. He could not say what it is.
Not in any specific case — for individual curves, careful computation can sometimes enumerate rational points. The gap was universal: there was no formula, no bound, no ceiling. Given a curve of genus five over the rationals, Faltings' theorem guarantees it has finitely many rational points, but that finite number could be ten or ten billion. The proof used compactness — the space of abelian varieties of bounded height is finite — and compactness arguments say “there are finitely many” without saying how many. The existence is global; the count is local; and the proof technique was purely global.
For four decades, mathematicians attacked the problem from multiple directions. Vojta gave an alternative proof in 1991 using Diophantine approximation, but his constants depended on the specific curve. Stoll, and later Katz-Rabinoff-Zureick-Brown, produced explicit bounds for restricted classes — curves whose Mordell-Weil rank was small relative to the genus — using p-adic analysis. These were genuine counts, but they worked only in what is called the Chabauty range, where rank is small enough for p-adic methods to constrain the points. Outside this range, the problem remained existential.
In 2021, Dimitrov, Gao, and Habegger proved a uniform bound exists — a single number depending only on the genus and the number field, not on the specific curve. This was the uniform Mordell-Lang conjecture, a qualitative advance. But the constant was proved to exist without being computed. It was known to be finite; it was not known to be any particular number.
In February 2026, Yu, Yuan, and Zhou made it explicit. Their paper, posted from the Toulouse Mathematics Institute and Peking University, provides the first computable formula for the uniform bound on rational points. The technique is what makes the result structurally interesting: they import Bergman kernel localization — a tool from complex differential geometry — into arithmetic geometry.
Bergman kernels are reproducing kernels for spaces of holomorphic sections on complex manifolds. They encode analytic capacity: how many independent holomorphic functions a manifold can support near a given point. Localization of Bergman kernels — understanding their asymptotic behavior in small neighborhoods — is a technique developed by Tian, Zelditch, and Catlin in the 1990s and 2000s. It is naturally explicit because it is naturally local. You estimate the kernel near a point, and the estimate gives you a number.
The bridge is Arakelov geometry, which sits between the algebraic world (where the curves live) and the analytic world (where the kernels live). Arakelov intersection theory assigns heights to points on arithmetic varieties by combining algebraic intersection numbers with analytic data — Green's functions, metrics on line bundles. The heights control how many rational points a curve can have. But the analytic data, traditionally computed through global methods, had always produced existence results rather than explicit estimates.
Yu, Yuan, and Zhou's insight is that Bergman kernel localization applied to the Arakelov Kähler forms gives explicit constants. The same objects that Faltings and Vojta used to prove finiteness contain, when analyzed locally rather than globally, enough quantitative information to produce a bound. The proof does not introduce new abstract machinery. It re-examines existing machinery with a technique that extracts numbers instead of existence claims.
The bound is almost certainly far from sharp. The record for rational points on a genus-two curve over the rationals is in the hundreds; the explicit universal bound is likely astronomically larger. But the bound exists, and it is computable, and the gap between Faltings and this result — from “finite” to “at most this many” — is the gap between knowing something is true and knowing how true it is. For forty-two years, the machinery that proved finiteness could not produce a count, because the proof techniques were global and global techniques do not count. The resolution required thinking locally, and the local thinking was always available in the analytic side of the same theory. The answer was embedded in the question, visible only from a different scale.