Elliptic Zagier Classes and Beilinson Regulators

Definitions and mathematical statement

The verified clean question is whether suitable nonzero motivic-cohomology classes of an elliptic curve have Beilinson regulators given by elliptic polylogarithms, as predicted by the elliptic Zagier conjecture.

An elliptic curve $E$ is a smooth projective genus-one curve with a chosen origin over a field $k$. For degrees $i,j$, write $H^i_{\mathcal M}(E,{\mathbb{Q}}(j))$ for rational motivic cohomology. After choosing an appropriate complex realization, the Beilinson regulator maps such classes to the corresponding Deligne or absolute-Hodge cohomology group. For example, with a fixed embedding $k\hookrightarrow{\mathbb{C}}$, its complex-realization form is \[ r_{\mathrm B}:H^i_{\mathcal M}(E,{\mathbb{Q}}(j)) \longrightarrow H^i_{\mathcal D}(E_{{\mathbb{C}}},{\mathbb{R}}(j)), \] with the additional real/Galois realization conventions chosen for the intended arithmetic setting.

The desired content is the existence or construction of specified classes $0\ne\xi$ whose regulator images equal the appropriate elliptic-polylogarithm expressions. The extract does not state the field of definition, cohomological degree and weight, admissible points or cycles, polylogarithm normalization, or an explicit regulator identity. It does not quantify "suitable" or specify whether the assertion is for every elliptic curve, every weight, or a restricted family. No particular $K$-group, regulator determinant, special $L$-value or nonvanishing of $r_{\mathrm B}(\xi)$ is therefore added. Nonzero motivic class and nonzero regulator image are distinct requirements.

The literature assessment checked on 21 August 2026 explicitly says this is an elliptic analogue and motivic programme without one isolated precise conjecture whose resolution can be verified from the extraction. That qualification remains despite the clean statement's corrected-and-verified label. The record provides a faithful programme-level target, not enough data for a unique fully quantified elliptic Zagier formula or a reliable current-status conclusion.

Short English statement

Construct the intended nonzero elliptic motivic classes and express their Beilinson regulators through elliptic polylogarithms. The source omits the degrees, admissible data and formula, so this remains a programme-level question rather than one specified regulator identity.

Sources

• Polylogarithms (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/48, pp. 2539--2590, https://doi.org/10.4171/owr/2004/48; proposed year supplied as 2004.
• The sole literature source in the assessment checked 21 August 2026 is the original report, https://doi.org/10.4171/owr/2004/48. No separate precise elliptic Zagier theorem or later resolution is cited.
• Local UnsolvedMath registry, permanent ID $30000144$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. The clean wording is labelled corrected and verified, while the attached assessment explicitly says the extracted programme lacks a uniquely isolated conjecture.