Local Nonvanishing in Arithmetic Gan–Gross–Prasad Theory

Definitions and mathematical statement

Scope of the supplied passage. The record for permanent UnsolvedMath ID $30005396$ contains a local nonvanishing assertion and a theorem giving cases of an unnamed conjecture. It does not state that conjecture. Consequently, a complete unresolved mathematical problem cannot be recovered from this record. The following preserves the supplied assertions, with the missing hypotheses explicitly identified.

Local notation and recorded assertion. Write $H_v=H(F_{0,v})$, and let $\pi_v^\vee$ denote the contragredient of the representation $\pi_v$ appearing in the excerpt. Here $\boxtimes$ denotes the external tensor product, and $\operatorname{Hom}_{H_v}(\pi_v,{\mathbb{C}})$ denotes the space of $H_v$-invariant linear functionals, with the trivial action on ${\mathbb{C}}$; the required representation category is not specified. The excerpt describes an Ichino--Ikeda canonical invariant linear functional \[ \alpha_v\in\operatorname{Hom}_{H_v\times H_v} (\pi_v\boxtimes\pi_v^\vee,{\mathbb{C}}) \] by the matrix-coefficient integral \[ \alpha_v(\phi_v,\phi_v^\vee) =\int_{H(F_{0,v})} \langle\pi_v(h)\phi_v,\phi_v^\vee\rangle_v\,{\,\mathrm{d}} h, \qquad \phi_v\in\pi_v,\quad\phi_v^\vee\in\pi_v^\vee, \] and records the equivalence \[ \alpha_v\ne0 \quad\Longleftrightarrow\quad \operatorname{Hom}_{H(F_{0,v})}(\pi_v,{\mathbb{C}})\ne0. \] The integral is a reconstruction of the damaged source notation. The ambient groups, the embedding giving the action of $H_v$, the pairing, the Haar measure, convergence or regularization, and any normalization factors are not supplied. These formulas therefore record the excerpt's assertion subject to its missing ambient hypotheses; they do not assert it for arbitrary groups and representations.

Recorded proved cases, with the conclusion missing. The passage labels the following assertion "Theorem 1": the unnamed conjecture holds under the following numbered conditions, presented together:

1. $p>2n$.
2. The extension $F/F_0$ is unramified everywhere, and all $2$-adic places are split in $F$.
3. Every place $v\mid p$ is split in $F$, and the corresponding $\pi_v$ are unramified and ordinary.
4. The representation $\pi$ is stable cuspidal. For the local factorization \[ \pi_v=\pi_{n,v}\boxtimes\pi_{n+1,v}, \] the following additional conditions are imposed:
   • If $v$ is split, at least one of $\pi_{n,v}$ and $\pi_{n+1,v}$ is unramified.
   • If $v$ is inert, each of these two factors is either unramified or almost unramified. The excerpt explains "almost unramified" as admitting a nonzero invariant vector under a parahoric subgroup defined by a vertex lattice of type $1$.

The ambient meanings and hypotheses for $F,F_0,p,n,H,\pi$, the precise notions of stability and ordinarity, and the range of places in condition (4) are absent. The excerpt also does not specify the scope of "unramified everywhere." The supplied curation associates the unnamed conjecture with a $p$-adic height conjecture in arithmetic Gan--Gross--Prasad theory, but gives neither its height identity nor its definitions and quantifiers. Thus the displayed hypotheses cannot be turned into a complete theorem or an unresolved conjecture from the supplied evidence alone.

Short English statement

The supplied passage records a local nonvanishing criterion and lists hypotheses under which an unstated $p$-adic height conjecture is said to hold. It supplies neither the conjecture itself nor the full ambient setup. Its supplied "partially solved" label is dated curation, while its statement-verification field marks the problem formulation as unrecoverable. This statement-only draft requires human mathematical and source review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30005396$, source problem code catalogue entry, title "Local Nonvanishing in Arithmetic Gan--Gross--Prasad Theory." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2023$; no proposer is identified.
• Supplied report citation: Arithmetic of Shimura Varieties (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/5, pp. 261--326. https://doi.org/10.4171/owr/2023/5. The background describes these reports as documenting mathematical workshops and their open-problem discussions. No precise page within this range is supplied for the excerpt.
• Supplied literature source: Disegni and Zhang, GGP cycles and p-adic L-functions. https://arxiv.org/abs/2410.08401. This is the attribution supplied by the record; the excerpt's reference "[1]" is not explicitly identified by a bibliography.
• Supplied literature source: Oberwolfach Report 5/2023. https://doi.org/10.4171/owr/2023/5.
• Literature assessment dated exactly 2026-08-22: the supplied status is "partially solved." The supplied assessment says that the recorded theorem gives proved cases of the $p$-adic height conjecture, including split ordinary and specified stable cuspidal situations, and does not give a general proof. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is "unrecoverable." Its statement-verification field reports that the official report gives a local nonvanishing fact followed by proved cases, without formulating an unresolved problem. This verification description is supplied metadata, not a claim of source consultation in preparing this draft.