Multiplicity Formula for Elliptic Arthur Parameters

Definitions and mathematical statement

The supplied record asks about parameters described as elliptic Arthur parameters of Ramanujan type. It uses an auxiliary symbol $\psi_N$, an assignment $\phi\mapsto\pi^{\psi_N}(\phi)$, coefficients $c_{\pi}^{\psi_N}$, and objects $S_\phi$, with $|S_\phi|$ denoting cardinality. The record does not define these objects or specify their ambient setting, so these descriptions preserve the source notation without supplying missing definitions.

The proposed identity is \[ c_{\pi^{\psi_N}(\phi)}^{\psi_N}=|S_\phi| \qquad\text{for every elliptic Arthur parameter $\phi$ of Ramanujan type.} \] The question is whether this identity holds for every such $\phi$ in the intended source setting. A fully specified mathematical formulation requires the underlying group and field, the local or global setting, the meaning of Ramanujan type and ellipticity, the admissible choice and quantification of $\psi_N$, the construction of $\pi^{\psi_N}(\phi)$, the definition and normalization of $c_{\pi}^{\psi_N}$, and the definition of $S_\phi$, including any quotient conventions and finiteness assumptions. None of these missing specifications is imposed here.

Short English statement

The question asks whether the coefficient attached to $\pi^{\psi_N}(\phi)$ equals $|S_\phi|$ for every elliptic Arthur parameter of Ramanujan type. The supplied record labels this claim open, but does not provide the definitions and hypotheses needed for a self-contained formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002588$, title: Multiplicity Formula for Elliptic Arthur Parameters; source problem code catalogue entry. The collection attributes its problems to open-problem discussions documented in Oberwolfach Reports. No individual proposer is supplied; the proposed year is recorded as 2014.
• Supplied report citation: Modular Forms (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/22, pp. 1221--1277. https://doi.org/10.4171/owr/2014/22. The report label gives 2015, while the DOI and recorded proposed year indicate 2014; this discrepancy is unresolved.
• Lapid--Mao, Whittaker--Fourier coefficients, as labeled in the supplied literature sources. https://doi.org/10.1016/j.jnt.2013.10.003.
• Lapid--Mao, metaplectic analogue, as labeled in the supplied literature sources. https://msp.org/ant/2017/11-3/ant-v11-n3-p07-s.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the displayed identity open in its stated generality and reports that available results establish only local or restricted global cases. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.