Denominators of Hodge integrals and Minkowski numbers

Abstract

We prove that the least common denominator of all genus $g$ Hodge integrals is the Minkowski number $T_{2g}$ or, equivalently, the Bhargava factorial $(2g+1)!_{\mathbb{P}}$ associated to the set of primes $\mathbb{P}\subset\mathbb{N}$. As a corollary, we prove a conjecture of Liu-Xu on the multiplicity of $3$ in the least common denominators of certain descendant integrals, and generalise their result on a conjecture of Itzykson-Zuber to Hodge integrals with $λ$ classes. Namely, the order of the automorphism group of any pointed stable curve of genus $g'\le g$ divides the least common denominator of all genus $g$ Hodge integrals. We also comment on the use of generative AI via IMProofBench, which played a fruitful part in the development of this (entirely human-written) paper.