Canonical Triple-Product $p$-Adic $L$-Values and Rational Points

Definitions and mathematical statement

The supplied record concerns symbols $p,E,f,g,h,\rho$, a triple-product $p$-adic $L$-function $L_p(f,g,h)$, and a twisted $L$-function $s\mapsto L(E,\rho,s)$. It does not specify the admissible data or their hypotheses. In particular, the objects $f,g,h$, the representation $\rho$, and the meaning of $E(\overline{{\mathbb{Q}}})_\rho$ are not defined in the supplied text. We retain this last notation as the source's designation for the space containing the proposed point; we do not choose an interpretation of its subscript.

A function $F(s)$ has a simple zero at $s=1$ if it is holomorphic in a neighborhood of $1$ and satisfies $F(1)=0$ and $F'(1)\ne 0$. Write $\log_E$ for the formal-group logarithm mentioned in the record. Its normalization, domain, and the choices needed to evaluate it on the proposed point are unspecified.

Conjectural target, subject to unresolved source qualifications. For the data and hypotheses intended by the source, there exists a canonically specified multiple $L_p^{\mathrm{can}}(f,g,h)$ of $L_p(f,g,h)$, together with a point described by the source as a rational point \[ P_\rho\in E(\overline{{\mathbb{Q}}})_\rho, \] such that \[ L_p^{\mathrm{can}}(f,g,h)(2,1,1)=\log_E(P_\rho) \] and \[ P_\rho\text{ is nontrivial} \quad\Longleftrightarrow\quad \left(s\mapsto L(E,\rho,s)\right)\text{ has a simple zero at }s=1. \] Here $(2,1,1)$ is the evaluation point appearing in the source; its parameter space is not supplied. The permitted multiplier and its canonical characterization are also unspecified. The terms "rational point" and "nontrivial" are retained without identifying a field of rationality or choosing between nonzero and non-torsion interpretations. Thus the displayed target records the supplied conjecture, but the record does not support a fully quantified formulation over a defined class of inputs.

Short English statement

The conjecture asks for a canonical normalization of a triple-product $p$-adic $L$-function whose value at $(2,1,1)$ is the formal-group logarithm of a point $P_\rho$. That point should be nontrivial exactly when $L(E,\rho,s)$ has a simple zero at $s=1$. The admissible data, normalization, point space, and meaning of nontriviality require clarification from the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004476$, exact source problem code catalogue entry, entitled "Canonical Triple-Product $p$-Adic $L$-Values and Rational Points," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2020 and names no individual proposer.
• Supplied source citation: Arithmetic Geometry (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/20, pp. 1023--1082. Original OWR report: https://doi.org/10.4171/owr/2020/20. The supplied assessment identifies the target as Conjecture 5; no more precise location is supplied.
• Supplied literature source: "Darmon--Rotger diagonal cycles II," https://web.mat.upc.edu/victor.rotger/docs/DR2.pdf.
• Supplied literature source: "p-adic families of diagonal cycles," https://arxiv.org/abs/2207.01314.
• Literature checked at: 2026-08-22. This is supplied dated curation, not an independently verified assessment. The record labels the problem partially solved and states that the Conjecture 5 value and nontriviality criterion remain unproved in general, while Darmon--Rotger establish new analytic-rank-zero cases and conditional higher-rank Selmer consequences. Its background characterizes this as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry assigns the statement status corrected_verified and reports a check against the official Oberwolfach report, asserting preservation of its hypotheses, notation, and target. This is a supplied verification claim; no independent source consultation or mathematical verification is asserted here.