Manin's conjecture for a quartic del Pezzo surface with A_4 singularity

Monstrous moonshine relates the representation of the Monster finite sporadic simple group to the distinguished modular functions, called Hauptmoduln. Chen-Yui~\cite{Chen-Yui} showed that the CM values of Hauptmoduln which appeare in monstrous moonshine (but not all) are algebraic integers, which is similar to the singular moduli of the $j$-function. In this paper, we generalize this result to Hauptmoduln whose $q$-coefficients are cyclotomic integers. A main idea for our proof is the use of generalized modular equations for Hauptmoduln, which was introduced by Cummins-Gannon~\cite{Cummins-Gannon} in the study of monstrous moonshine. As an application, we show that if a formal $q$-series satisfies the special combinatoric property called complete replicability, its CM values are algebraic integers, without assuming the modular invariance.

math.NT↗

arXiv2026

An interpolation of Bradley's sum formula

The sum formula for $q$-multiple zeta values is a well-known relation. In this paper, we present its generalization for the $q$-multiple zeta function.

math.NT↗

arXiv2026

On Graphical Partitions with Restricted Parts

An integer partition of $n$ is called graphical if its parts form a degree sequence of a simple graph. While unrestricted graphical partitions have been extensively studied, much less is known when the parts are restricted to a prescribed set. In this work, we investigate the probability that a uniformly random partition of an even integer $n$, subject to such restrictions, is graphical. We establish an upper bound on this probability expressed solely in terms of the Durfee square of the partition. Additionally, we evaluate the most-probable size of the Durfee square, depending solely on $n$ and the restrictions on the parts. Thus, we establish typical bounds on the probability that such random partitions are graphical. Our approach employs the Nash-Williams graphical condition, the saddle-point method and Edgeworth expansions.

math.NT↗