$p$-Adic Regulators for Triple-Product $L$-Functions

Definitions and mathematical statement

The supplied record concerns the indefinite triple-product setting and a $p$-adic $L$-function denoted by $L_p(f,g,h)$, with distinguished parameter value $(2,1,1)$. Here $p$ denotes a prime. The record does not specify the admissible primes, the forms $f,g,h$, their varying families, the parameter domain, the meaning of "indefinite," or the normalization of the $L$-function. These are unresolved parts of the formulation.

Conjectural target recorded by the source. Formulate and prove, under explicit hypotheses in this setting, a Bloch--Kato-style law expressing the behavior of $L_p(f,g,h)$ near $(2,1,1)$ in terms of a $p$-adic regulator, as a generalization of the Birch--Swinnerton-Dyer expectation. The supplied record does not identify the relevant arithmetic representation, Selmer group, regulator construction, order of vanishing, leading coefficient, or normalization factors. Consequently it does not determine a precise regulator identity or a quantified conjecture.

Restricted formula reported in the record. The original statement displays \[ \left.\partial_k^2 L_p(f,g,h)\right|_{(2,1,1)} =\log_p(P_\phi)\,\log_p(P_\psi). \] The notation $\partial_k^2$ denotes the second derivative in the parameter called $k$ by the source. The record does not define that parameter, the objects $P_\phi,P_\psi$, their labels $\phi,\psi$, or the map denoted by $\log_p$. Its supplied literature assessment describes this formula as a proved theorem under explicit restricted hypotheses, but those hypotheses and the normalization conventions are absent from the supplied record. Thus the displayed equality is retained as a source-reported restricted formula, not asserted here for every indefinite triple product. The broader regulator expectation is described by the supplied assessment as conjectural.

Short English statement

The problem asks for a Bloch--Kato-style $p$-adic regulator law governing a triple-product $L$-function near $(2,1,1)$. The supplied record reports a restricted second-derivative formula, but omits the hypotheses and definitions needed to state either that theorem or the broader conjecture precisely.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004475$, title "$p$-Adic Regulators for Triple-Product $L$-Functions," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2020$ and no named proposer.
• Supplied report citation: Arithmetic Geometry (2021), Oberwolfach Reports, pp. 1023--1082, DOI 10.4171/owr/2020/20. Original OWR report: https://doi.org/10.4171/owr/2020/20.
• Supplied literature reference: Darmon--Rotger, "diagonal cycles I," https://doi.org/10.24033/asens.2227.
• Supplied literature reference: Darmon--Rotger, "diagonal cycles II," https://web.mat.upc.edu/victor.rotger/docs/DR2.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, describes the restricted second-derivative formula as proved under explicit hypotheses, and describes the broader $p$-adic Bloch--Kato or Birch--Swinnerton-Dyer expectation as conjectural. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports checking it against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft.