Arithmetic Intersection Formula for Special Cycles

Definitions and mathematical statement

Let $n,L,\rho,U$ denote the auxiliary data intended in Conjecture 0.2 of the cited report. The supplied record does not specify their admissible values or hypotheses. Retain the source notation $H_L^{1-n/2,\bar\rho}$ for the space containing $f$, and write $c_f^+(0,0)$ for its distinguished constant coefficient, denoted $c^+(0,0)$ in the record. Let $\widehat Z(f)$, $Z(U)$, $\langle\ ,\ \rangle_{\mathrm{Fal}}$, $\deg Z(U)$, and $\xi(f)$ denote, respectively, the source's arithmetic cycle associated with $f$, cycle associated with $U$, Faltings pairing, cycle degree, and image under the map $\xi$. These objects and their normalizations are not defined in the supplied record. Interpret the prime in $L'(\xi(f),U,0)$ as differentiation of $L(\xi(f),U,s)$ with respect to $s$ at $s=0$.

The proposed statement is that, for every $f$ in the source's intended space satisfying the stated constant-term condition, \[ f\in H_L^{1-n/2,\bar\rho},\qquad c_f^+(0,0)=0 \quad\Longrightarrow\quad \langle\widehat Z(f),Z(U)\rangle_{\mathrm{Fal}} =\frac{\deg Z(U)}{2}\,L'(\xi(f),U,0). \] This is a statement-only transcription relative to the intended source setup. A self-contained conjecture requires the missing definitions, admissibility hypotheses, and normalization conventions from the cited report; the supplied record alone does not determine them.

Short English statement

The conjecture asks whether the Faltings pairing of the arithmetic cycle attached to every admissible $f$ with $Z(U)$ equals one half of $\deg Z(U)$ times the indicated $L$-function derivative, provided $c_f^+(0,0)=0$. The auxiliary setup and normalization conventions remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001273$, title Arithmetic Intersection Formula for Special Cycles, exact source problem code catalogue entry, Oberwolfach Reports Open Problems collection, proposed year 2009. No individual proposer is supplied.
• Algebraische Zahlentheorie (2009), Oberwolfach Reports, pp. 1667--1730, DOI 10.4171/owr/2009/30. https://doi.org/10.4171/owr/2009/30. The supplied verification note identifies Conjecture 0.2 and says that the arithmetic intersection formula and vanishing-constant-term hypothesis were checked against the official report. This is supplied verification metadata, not an independent verification here.
• Supplied literature reference, labelled "big CM heights": https://arxiv.org/abs/1008.1669.
• Supplied literature reference, labelled "Kudla--Rapoport intersections": https://annals.math.princeton.edu/2012/176-2/p09.
• Supplied literature reference, labelled "original OWR report": https://doi.org/10.4171/owr/2009/30.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that the report proves the $n=1$ Gross--Zagier specialization and several low-dimensional cases, that later arithmetic-intersection work extends the formula, and that the full stated conjecture is not generally settled. This is supplied dated curation, not an independently verified literature assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.