Points from symmetries

0:00Points from symmetries

Take a curve of genus 2. Over the complex numbers, it’s a surface with two holes. In 1983, Gerd Faltings proved that such a curve has only finitely many rational points. But his proof doesn’t tell you where they are. That same year, Alexander Grothendieck wrote to Faltings with a radical idea. The points might be recorded entirely in group theory: in the curve’s fundamental group, its loops, together with the symmetries of the algebraic numbers. This is the section conjecture. Result 019 in OpenAI’s math repository claims to prove its p-adic version, and, with known descent results, the full conjecture for infinitely many modular curves over the rationals.

0:46Sections

Two groups meet here. Delta is the geometric fundamental group: loops on the curve, seen over the algebraic numbers. G is the Galois group: the symmetries of the algebraic numbers. Together they form a bigger group, Pi, the arithmetic fundamental group, which maps onto G, with Delta as its kernel. A section is a way to lift G back into Pi: a homomorphism s from G to Pi, so that following s by the projection gives back every element of G. Each rational point gives one. A rational point is a map from a single point into the curve, and maps induce maps on fundamental groups. Grothendieck conjectured the converse, for curves of genus at least 2. Every section comes from a rational point, and different points give different sections, counted up to conjugation by Delta. Rational points would become pure group theory. A picture helps. The Galois symmetries act on the curve, and the rational points are exactly the points they fix. A section is a weaker thing, a point that’s fixed only up to deformation. The conjecture says that on these curves, nothing is fixed up to deformation without being truly fixed.

1:59The p-adic version

There’s a local version. Replace the rationals by the p-adic numbers, or a finite extension of them. Now a curve can have infinitely many points, or none at all, but the question still makes sense: is the map from points to sections a bijection? Different points giving different sections is the easier half. The hard half is showing that every section comes from a point. In 2005, Jochen Koenigsmann proved a birational version, using the Galois group of the curve’s function field. In 2017, Florian Pop and Jakob Stix showed that every section singles out a p-adic valuation on the curve. In the p-adic world, a curve looks like a branching, tree-like space. Rational points sit at tips of its branches, and other valuations fill in the rest. Pop and Stix placed every section somewhere in this space. The missing step was to show it always sits at a rational point.

2:55The claim

The main preprint, 24 pages, dated October 6, claims exactly that. For every curve of genus at least 2, over every finite extension of the p-adic numbers, points and sections match one to one. A 7-page companion, dated September 24, not used in the main proof, constructs étale covers with a prescribed sheet outside a few small disks. Then the global consequence. Combined with earlier descent results of Michael Stoll, David Harari and Jakob Stix, the section conjecture follows over a number field whenever a finite descent condition holds. That includes the modular curves X₀(N) and X₁(N) of genus at least 2 over the rationals. There are infinitely many, starting with X₀(22), and X₁(13).

3:44How the proof goes

The heart of the proof is a statement about approximation. Build the field of all étale covers of the curve, the covers with no branch points. Complete it at any p-adic absolute value. The claim is that this completion already contains every finite extension of the curve’s function field, branched or not. That’s enough. A section at such a valuation then lifts to the Galois group of the whole function field, where Koenigsmann’s theorem pins it to a rational point. A valuation that isn’t a point would leave one field with two incompatible valuations, and a classical principle of Friedrich Karl Schmidt rules that out. Why can étale covers approximate branched ones? Near a branch point, a branched cover looks like t = zᵉ. If some other cover supplies an e-th root of t on the whole ring around the branch point, the cover splits there into separate sheets. Cut out the small disk, glue in flat sheets, and you get an étale cover that, outside the disk, lies over the original. The roots come in two stages. First, roots of a fixed odd prime order ℓ, other than p, from the curve’s stable reduction, using a lifting theorem from tropical geometry and the resolution of nonsingularities of Mochizuki and Tsujimura. Then roots of every order, from covers of the projective line whose skeleton has two vertices joined by ℓ edges, the p-adic uniformization of their Jacobians, and the p-adic logarithm.

5:14Where it stands

There’s no Lean proof for this result. These are preprints, not peer reviewed: 24 pages and 7. The proof leans on many recent results, including the resolution of nonsingularities of Mochizuki and Tsujimura, which the paper cites as a 2024 preprint. And the global conjecture stays open in general: the descent condition is itself only conjectured for most curves. Grothendieck guessed that symmetries alone could find the points. If the proof holds, over the p-adic numbers, they always do.