we-solve (beta)

Math-2810 · mathematics · Number theory

Finite étale covers with a prescribed exterior sheet and p-divisible disk degrees

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix a prime p, a finite extension $k/Q_p$, its algebraic closure and completed algebraic closure Omega. Let X over the algebraic closure be a smooth proper connected curve of genus at least two. Choose finitely many disjoint open disks $V_i$ of positive value-group radii strictly inside rational parameter charts, so that $U=X^an$ minus their union is nonempty and contains an Omega-rational point. Does one smooth proper connected curve Y admit a finite etale surjective morphism to X with a component over U mapping isomorphically to U and every component over every prescribed $V_i$ having degree divisible by p, without shrinking the disks?

Short English statement

Fix a prime p, a finite extension $k/Q_p$, its algebraic closure and completed algebraic closure Omega. Let X over the algebraic closure be a smooth proper connected curve of genus at least two. Choose finitely many disjoint open disks $V_i$ of positive value-group radii strictly inside rational parameter charts, so that $U=X^an$ minus their union is nonempty and contains an Omega-rational point. Does one smooth proper connected curve Y admit a finite etale surjective morphism to X with a component over U mapping isomorphically to U and every component over every prescribed $V_i$ having degree divisible by p, without shrinking the disks?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2810-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2810

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 019; 2026-09-24).

Finite étale covers with a prescribed exterior sheet and p-divisible disk degrees

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 019; 2026-09-24), the asserted result is that prescribed disjoint small disks on a proper hyperbolic $p$-adic curve admit one connected finite étale cover with a sheet over their entire exterior and degree divisible by $p$ on every disk component.

Mathematical statement of the claimed result

Manuscript 1.
Fix a prime $p$, a finite extension $k/\mathbb Q_p$ and its algebraic closure $\bar k$, with analytic completion $\Omega=\widehat{\bar k}$. Let $X/\bar k$ be smooth proper connected of genus at least two. At distinct $b_1,\ldots,b_d$, $d\ge1$, choose disjoint parameter disks $B_i$ with coordinates $t_i$ and radii $0<r_i<R_i$ in $|\Omega^{\times}|$. Put $V_i=\{|t_i|<r_i\}$ and require the compact exterior $U=X^{\rm an}\setminus\bigcup_iV_i$ nonempty and containing an $\Omega$-rational point. The claim gives a smooth proper connected curve $Y/\bar k$ and finite étale surjection $g:Y\to X$ such that \[ \exists C\text{ component of }g^{-1}(U):\ C\simeq U, \qquad p\mid\deg(D\to V_i)\quad\text{for every component }D\subseteq g^{-1}(V_i). \] All inverse images are analytic over $\Omega$, and the prescribed disks are retained without shrinking.

Outline of the argument

For manuscript 1 (family 019; 2026-09-24), resolution covers first create positive-genus skeletal vertices, and auxiliary torsors force a cycle over each disk. Degree-$p$ graph characters split over the exterior; a quotient by an exterior stabilizer keeps one exterior sheet while retaining the required disk degrees.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 019; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →