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Milne’s rationality conjecture

Compatible cohomological realizations alone do not establish rational intersection numbers with new divisor classes arising after reduction.

Progress records 1

Understand the problem ↓

1 record · How we rate progress

Understanding the problem

Read the background

Compatible cohomological realizations alone do not establish rational intersection numbers with new divisor classes arising after reduction. Milne’s conjecture asks for that additional rationality.

The statement concerns good reduction. The algebraic-cycle specialization conclusion invokes the separate CM Hodge theorem; this is not a proof of the unrestricted Hodge conjecture.

Why it matters

If correct, resolves the stated rationality problem across residue characteristics, beyond compatibility of realizations.

What counts as progress

A sourced resolution or substantial advance on the precise target, with hypotheses, prior context, and verification evidence stated separately.

What to watch for

  • Assess the precise statement and assumptions before comparing results.
  • The statement concerns good reduction. The algebraic-cycle specialization conclusion invokes the separate CM Hodge theorem; this is not a proof of the unrestricted Hodge conjecture.
  • Track independent assessments, dependencies, corrections, and withdrawals separately from importance.

Background review: 2026-10-11. Selected sources:

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