Automorphic L-Functions from Sigma–Rho Poisson Summation

Definitions and mathematical statement

Write $L(s,\sigma,\rho)$ for the automorphic $L$-function named in the supplied record, with complex variable $s$. The record does not specify the underlying global field, the group, the class of automorphic representations $\sigma$, the representation $\rho$, or the normalization of this $L$-function.

Let $\mathcal P_{\sigma,\rho}$ denote the conjectural $(\sigma,\rho)$-Poisson summation formula referred to in the source. This notation is a placeholder: the supplied record does not state that formula, its test-function space, its transform, its convergence conditions, or any correction terms. Likewise, let $\mathcal{FE}_{\sigma,\rho}$ denote the intended functional equation, whose precise form is not supplied.

The requested implication is \[ \mathcal P_{\sigma,\rho} \quad\Longrightarrow\quad \left\{ \begin{array}{l} L(s,\sigma,\rho)\text{ admits meromorphic continuation},\\ L(s,\sigma,\rho)\text{ satisfies }\mathcal{FE}_{\sigma,\rho}. \end{array} \right. \] It is intended to apply to the pairs $(\sigma,\rho)$ covered by the conjecture. However, neither that quantified class nor the initial domain and target domain of continuation are specified in the supplied record. The record also does not distinguish a completed $L$-function from a partial or finite-part $L$-function, or specify the dual data, normalization, and factors in the functional equation. Consequently, this is a conditional statement schema requiring those data before it becomes a fully specified mathematical assertion.

The supplied background additionally discusses an established $\mathrm{GL}_n$ identity, refined Poisson formulas for selected Braverman--Kazhdan spaces, and an expected direct proof for all split classical groups with the standard representation. These are supplied contextual qualifications; the record does not explain whether the last scope restricts the displayed implication, concerns proving its antecedent, or describes a separate extension.

Short English statement

Show that the conjectural $(\sigma,\rho)$-Poisson summation formula yields meromorphic continuation and the appropriate functional equation for $L(s,\sigma,\rho)$. The supplied record leaves the admissible data and both formulas unspecified, so the statement remains a conditional schema requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004786$, title Automorphic L-Functions from Sigma--Rho Poisson Summation, collection Oberwolfach Reports Open Problems, exact source code catalogue entry. The record gives proposed year 2021 and names no proposer.
• Supplied source citation: Automorphic Forms, Geometry and Arithmetic (2022), Oberwolfach Reports, pp. 2089--2156, DOI 10.4171/owr/2021/39. https://doi.org/10.4171/owr/2021/39. The literature-source list labels this reference Oberwolfach Report 39/2021.
• Supplied literature reference: A refined Poisson summation formula. https://arxiv.org/abs/1707.06091.
• Supplied literature reference: Harmonic analysis on certain spherical varieties. https://arxiv.org/abs/2103.10261.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the $\mathrm{GL}_n$ identity is established and refined Poisson formulas are known for selected Braverman--Kazhdan spaces, and describes the proposed direct proof for all split classical groups with the standard representation as an expected extension rather than a general theorem. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that its standalone formulation and hypotheses were checked against the official Oberwolfach report. This verification claim is supplied metadata; the report and cited papers have not been independently consulted for this draft.