The local p-adic section conjecture and global consequences

1 · The essentials

This family of manuscripts studies how to recognize points on a curve whose coordinates belong to a chosen number field. Its starting point is the $p$ -adic numbers, which provide another way to measure how close numbers are. For the curves under consideration, the second manuscript claims an exact correspondence between these points and certain symmetry data attached to the curve. The curves must be smooth, proper, geometrically connected, and of genus at least two: these are essential conditions of the result. The first manuscript constructs special covers, meaning curves lying above the original curve with multiple sheets. The second also announces a consequence over number fields under an additional condition, including certain modular curves over the rationals. These results are not formalized in Lean and have not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a map covered with several transparent sheets, with a few small circular regions marked out. The first manuscript’s construction preserves a sheet that exactly reproduces the entire exterior of those regions, while requiring the number of sheets in each component above each region to be divisible by $p$ . This image captures neither $p$ -adic geometry nor the symmetry data in the second manuscript.

Why it matters

The claimed correspondence would characterize rational points on the curves concerned through their arithmetic fundamental group. The global consequence connects this local result to established finite-descent theorems. It remains conditional over number fields in general, but the abstract explicitly includes the modular curves $X_{0} (N)$ and $X_{1} (N)$ of genus at least two over $Q$ .

2 · Going further

Let $K$ be a finite extension of $Q_{p}$ , and let $X / K$ be a smooth proper geometrically connected curve of genus at least two. The local section conjecture asks for a bijection between the points of $X (K)$ and conjugacy classes of sections of the full arithmetic étale fundamental group of $X$ . A section splits the projection from this group to the Galois group of the base field. The second manuscript claims this bijection for every curve satisfying these hypotheses.

The first manuscript works over an algebraic closure of a $p$ -adic local field. Given finitely many disjoint small open disks, it constructs one connected finite étale cover satisfying all the requirements simultaneously: a sheet is isomorphic to the entire exterior of the disks, and the degree on every connected component above each disk is divisible by $p$ . The construction idea described in the abstract is thus to control both the exterior and the behavior above the disks at once. The abstracts do not give the steps connecting this construction to the section conjecture.

The second manuscript states the local result and then combines it with established finite-descent theorems. Over a number field, the announced consequence applies when the finite-cover descent locus equals the set of rational points. The abstract says this includes $X_{0} (N)$ and $X_{1} (N)$ of genus at least two over $Q$ ; it does not claim the global conjecture for all curves over all number fields.

Useful notions

$p$ -adic fields and number fields
Algebraic curves and rational points
Finite étale covers
Galois groups and sections
Finite descent

What was known before / what is new

Before

The second abstract mentions established finite-descent theorems, which are used to pass from the local result to the global consequence. The source presents the statements under study as the local $p$ -adic section conjecture and Grothendieck’s section conjecture. It supplies no further history or inventory of previously known cases.

New

The first manuscript announces a connected cover combining a prescribed exterior sheet with degree divisibility by $p$ above each disk. The second claims the local section conjecture for all curves satisfying the stated hypotheses over every finite extension of $Q_{p}$ . It derives a global consequence under a descent hypothesis, with the modular cases explicitly listed.

Limits and assumptions

The local result concerns only smooth proper geometrically connected curves of genus at least two over a finite extension of $Q_{p}$ , and uses the full arithmetic étale fundamental group. The cover construction takes place over an algebraic closure and assumes finitely many disjoint small open disks. The consequence over number fields requires equality between the finite-cover descent locus and the rational points. The abstracts do not explain the detailed passage from the construction to the local result, and do not settle the global case without this condition. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Rational point: A point on a curve defined over the chosen base field, which need not be $Q$ .
Finite étale cover: A finite algebraic map that behaves as a cover without ramification.
Arithmetic étale fundamental group: A group encoding a curve’s finite étale covers while taking its field of definition into account.
Section: A homomorphism providing a right inverse to the projection from the arithmetic fundamental group to the Galois group of the base field.
Conjugacy class: A class grouping sections identified by the conjugation action considered in the conjecture.

Keywords : Section conjecture · $p$-adic curves · Finite descent

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚ_p correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves $X_{0} (N)$ and $X_{1} (N)$ of genus at least two.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Étale covers with a prescribed exterior sheet

24/09/2026

To be verified

Let X be a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.

PDF GitHub folder

The p-adic section conjecture

06/10/2026

To be verified

We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚ_p. Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes $X_{0} (N)$ and $X_{1} (N)$ of genus at least two over ℚ.

PDF GitHub folder