Hodge Integrals over Genus-One Admissible Covers

Definitions and mathematical statement

For integers $d\geq 2$ and $n\geq 1$, write \[ \mathcal A_{n,d}=\operatorname{Adm}_{1}^{n+1}\left((2)^{2n}\right)_d \] for the source's notation for a moduli space of degree-$d$ admissible covers of a genus-one curve with $2n$ simple branch points and domain of genus $n+1$. The supplied text explicitly explains this interpretation for $d=2$; its higher-degree question uses the parallel notation. On the smooth connected locus, with no other ramification, the genus follows from Riemann--Hurwitz: \[ 2g(C)-2=d\left(2g(E)-2\right)+2n=2n. \] Connectedness, the treatment of boundary covers, and the precise moduli convention must be checked in the original source; they are not fully specified in the supplied extraction.

Under the usual interpretation, let $\mathbb E$ be the rank-$n+1$ Hodge bundle of the domain curve, whose fiber at a smooth domain $C$ is $H^0(C,\omega_C)$, and put $\lambda_j=c_j(\mathbb E)$, with $\lambda_0=1$. The interleaved source extraction appears to display the degree-two identity \[ \sum_{n=1}^{\infty}\frac{u^{2n-1}}{(2n-1)!} \int_{\mathcal A_{n,2}}\lambda_{n+1}\lambda_{n-1} =\frac{i}{24}\,\frac{1-e^{iu}}{1+e^{iu}}, \qquad i^2=-1. \] This is a reconstruction of the supplied display, whose transcription and normalization require source review. The source attributes this example to joint work with H.-H. Tseng.

The natural reconstruction of Question 2 is: for every integer $d>2$, determine a closed formula for \[ F_d(u)=\sum_{n=1}^{\infty}\frac{u^{2n-1}}{(2n-1)!} \int_{\mathcal A_{n,d}}\lambda_{n+1}\lambda_{n-1}, \] and identify the rational function $R_d$ satisfying \[ F_d(u)=R_d(q),\qquad q=-e^{iu}. \] The source asserts that the parallel higher-degree expression is rational in $q$. This assertion is supplied source information, not independently verified here. The displayed definition of $F_d$ interprets "parallel integral" by retaining the degree-two Hodge classes and generating-series weights; the extracted question does not independently specify these choices for higher degree.

A completely precise formulation remains conditional on identifying the intended admissible-cover stack, its compactification and integration class, branch-point labeling and automorphism weights, whether the genus-one target varies, and the coefficient field and intended expansion or analytic meaning of the rational-function identity. These choices must not be inferred from the notation alone.

Short English statement

For each integer $d>2$, the question asks for the explicit rational function of $q=-e^{iu}$ represented by the higher-degree analogue of a supplied degree-two Hodge-integral series. The supplied record labels the problem open and asserts rationality, but the extracted text leaves the precise moduli and normalization conventions unresolved. This is a statement-only reconstruction requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005558$, source problem code catalogue entry, title "Hodge Integrals over Genus-One Admissible Covers," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2023 and no explicit proposer in its proposer field.
• Recent Trends in Algebraic Geometry (2023), Oberwolfach Reports, pp. 1485--1542, DOI 10.4171/owr/2023/27. Supplied report reference: https://doi.org/10.4171/owr/2023/27.
• Supplied literature-source attribution: Pandharipande, "Maps to a moving elliptic curve," https://doi.org/10.4171/owr/2023/27. The supplied extract attributes the degree-two example to joint work with H.-H. Tseng.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports that the higher-degree integral is known to be rational in $q$, and states that no authoritative general formula resolving the extracted question was located. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels its statement corrected_verified and reports reconstruction and checking against the official report. These are supplied verification claims; no independent source visit or verification is asserted in this draft.