Arthur Parameters from Automorphic Descent Integrals

Definitions and mathematical statement

Write $\mathbb A$ for the adelic ring appearing in the source; its underlying global field is not specified. Let $\sigma$ and $\pi$ be irreducible cuspidal automorphic representations of \[ \operatorname{SO}_{2k}(\mathbb A) \quad\text{and}\quad \operatorname{SO}_{2ab+2k}(\mathbb A), \] respectively, as stated in the supplied record. Denote the source's proposed global Arthur parameter $\psi_{\operatorname{SO}_{2k}}$ by $\Psi$. The notation $\Psi\oplus(\tau,b)$ denotes the formal direct sum with the Arthur summand written $(\tau,b)$ in the source. The superscript $\psi$ in the descent integral is retained separately; the record does not identify it with $\Psi$.

Source question. Suppose that the specified automorphic descent integral satisfies \[ I_{a,k,l}^{\psi}(\tau,\sigma;\pi)\ne 0, \] and impose the parity conditions \[ \begin{cases} b\in 2\mathbb Z,&\text{if $\tau$ is of symplectic type},\\ b\in 2\mathbb Z+1,&\text{if $\tau$ is of orthogonal type}. \end{cases} \] Under the intended hypotheses of the source, is it true that \[ \left[\sigma\text{ has global Arthur parameter }\Psi\right] \quad\Longleftrightarrow\quad \left[\pi\text{ has global Arthur parameter }\Psi\oplus(\tau,b)\right]? \]

Unresolved qualifications. This records the proposed equivalence, but the supplied information does not determine a fully specified conjecture. It omits the global field, the forms of the special orthogonal groups, the domain and hypotheses on $\tau$, the ranges and relations of $a,b,k,l$, the datum $\psi$, and the definition and nonvanishing convention of the integral. The symplectic and orthogonal types therefore cannot be specified from the supplied record alone. The quantification over $\Psi$ and the conventions for global Arthur parameters of even special orthogonal groups also require clarification.

There is a possible dimension discrepancy that must remain unresolved. Under the conventional interpretation that $\tau$ is a representation of $\operatorname{GL}_{d}(\mathbb A)$, the block $(\tau,b)$ has standard dimension $db$. Compatibility with the displayed target group would consequently require \[ 2k+db=2ab+2k. \] The record does not specify $d$. In particular, an interpretation with $d=a$ would generally conflict with the displayed group dimension. Neither the group subscript nor the Arthur summand has been altered here.

Short English statement

The source asks whether nonvanishing of a specified automorphic descent integral forces an exact equivalence between the Arthur parameter of $\sigma$ and the parameter obtained for $\pi$ by adjoining $(\tau,b)$, with the stated parity restriction on $b$. Essential descent data and parameter conventions are missing, and the displayed dimensions may be incompatible under some interpretations of $\tau$. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001741$, title Arthur Parameters from Automorphic Descent Integrals, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2011 and names no individual proposer.
• Supplied report citation: Automorphic Forms: New Directions (2011), Oberwolfach Reports, pp. 719--779, DOI 10.4171/owr/2011/14. Source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2011/14. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or integral definition is supplied.
• Supplied literature-source link labelled "local descent": https://doi.org/10.1353/ajm.2012.0023. The record supplies no authors, title, theorem number, or precise account of which cases this reference establishes.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that the parameter relation is proved for tempered $\sigma$ and the "first $b$ cases", while the all-$b$ conjecture remains beyond those cases. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying verification note says that the mathematical content was checked against the official report and that notation, context, and question form were normalized without changing the source-level claim. That verification is reported provenance, not verification performed for this draft.