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Math-2406 · mathematics · Number theory

The mod-$\ell$ Bogomolov–Pop conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $K/k$ and $L/l$ be function fields of transcendence degree at least two over algebraically closed fields of characteristic different from $\ell$. Does the natural map from field isomorphisms, modulo Frobenius in positive characteristic, to Milnor-$K$-theory-preserving isomorphisms $K^*/(K^*)^\ell\to L^*/(L^*)^\ell$ give a bijection?

Formulation references: [S1], [S2].

Short English statement

Let $K/k$ and $L/l$ be function fields of transcendence degree at least two over algebraically closed fields of characteristic different from $\ell$. Does the natural map from field isomorphisms, modulo Frobenius in positive characteristic, to Milnor-$K$-theory-preserving isomorphisms $K^*/(K^*)^\ell\to L^*/(L^*)^\ell$ give a bijection?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2406-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2406

The reconstruction correspondence identifies compatible Milnor maps only after quotienting by one common nonzero scalar. That quotient is absent from the displayed target, so the exact formulation is not covered as written; at prime two the scalar ambiguity happens to be trivial.

The mod-\$
\textbackslash{}
ell\$ Bogomolov–Pop conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 009; 2026-10-05), the asserted result is that degree-one and degree-two Milnor $K$-theory modulo a prime different from the field characteristic recover the perfect closure of a function field of transcendence degree at least two, with a common scalar and Frobenius ambiguity.

Mathematical statement of the claimed result

Manuscript 1.
Fix any prime $\ell$, and for fields of characteristic different from $\ell$ put $V_F=F^{\times}/(F^{\times})^{\ell}$ and $R_F=\operatorname{span}_{\mathbb F_{\ell}}\{[x]\otimes[1-x]:x\ne0,1\}$. Compatible linear isomorphisms carry $R_K$ onto $R_L$. For finitely generated $K/k,L/l$ with algebraically closed constants and transcendence degrees at least two, the claim is \[ \operatorname{Isom}_{\rm F}^{i}(K,L)\xrightarrow{\sim} \operatorname{Isom}_{\rm M}(V_K,V_L)/\mathbb F_{\ell}^{\times}. \] The left side uses isomorphisms of perfect closures carrying $k$ to $l$, identified in positive characteristic under postcomposition by any integral power of Frobenius; in characteristic zero there is no such quotient. If characteristics differ it is empty. Thus a compatible mod-$\ell$ invariant isomorphism is claimed to force equal characteristics and relative transcendence degrees. Perfect closures and the Frobenius ambiguity are essential here.

Outline of the argument

For manuscript 1 (family 009; 2026-10-05), alternating characters first identify valuation data and residual curves. Bounds on the divisors occurring in pencils recover one-variable subfields; simultaneous value incidence then reconstructs the field and shows that all local scalar choices agree.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 009; 2026-10-05).

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

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