Beilinson's Regulator Conjecture

Definitions and mathematical statement

For a real vector space $V$, a ${\mathbb{Q}}$-structure means a ${\mathbb{Q}}$-linear subspace $W\subseteq V$ such that the natural map \[ W\otimes_{{\mathbb{Q}}}{\mathbb{R}}\longrightarrow V,\qquad w\otimes a\longmapsto aw, \] is an isomorphism. This definition explains the terminology when real Deligne cohomology is intended; the supplied record does not specify the coefficient convention for its Deligne cohomology.

Source-level conjectural statement. Let $X$ be a variety over ${\mathbb{Q}}$, and let $m,i$ be the cohomological parameters in the source, satisfying $m<2i$. The supplied record attributes to Beilinson's conjecture the prediction that the regulator carries an appropriate integral subspace of motivic cohomology to a ${\mathbb{Q}}$-structure on the relevant Deligne cohomology, and that the determinant of this regulator is given by the predicted special value.

Unresolved qualifications. This is a schematic conjectural target, not a fully specified mathematical assertion. The record does not specify the permitted class of varieties, the admissible ranges of $m,i$ beyond $m<2i$, the cohomological degrees and twists of the source and target, or the regulator itself. It does not define the integral subspace, including whether this means a lattice or a rational subspace selected by an integral construction. It also omits the relevant $L$-function, evaluation point, possible leading-term convention, determinant-line identifications, period factors, and normalization or equivalence relation for the proposed determinant equality. These data must be recovered before an exact regulator formula can be stated; no particular choices are made here.

Short English statement

The supplied record describes a Beilinson-type prediction relating an integral part of motivic cohomology, a rational structure on Deligne cohomology, and a special-value formula for a regulator determinant. Essential hypotheses and normalizations are missing, so the draft preserves the conjectural target while leaving its exact formulation unresolved. The supplied open-status label is not independently verified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003877$, exact source problem code catalogue entry, title "Beilinson's Regulator Conjecture," in the Oberwolfach Reports Open Problems collection. The record attributes the prediction to Beilinson's conjecture; no individual proposer is supplied. The proposed year is recorded as 2018.
• Supplied source citation: Algebraische Zahlentheorie (2019), Oberwolfach Reports, pp. 1785--1844, DOI 10.4171/owr/2018/30. https://doi.org/10.4171/owr/2018/30. The supplied citation's 2019 year and the report's 2018 identifiers are retained without reconciliation. No exact passage or problem page is supplied.
• Supplied literature source: "Oberwolfach Report 30/2018." https://ems.press/content/serial-article-files/46751.
• Supplied literature assessment, dated exactly 2026-08-21: the extracted regulator-formula claim is truncated and does not identify a settled theorem; the surrounding Beilinson-type conjectural statements are described as unresolved. The supplied status is "open." This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft, and it does not resolve the missing formulation data identified above.