Quasimodular moonshine and arithmetic connections

Abstract

We prove the existence of a module for the largest Mathieu group, whose trace functions are weight two quasimodular forms. Restricting to the subgroup fixing a point, we see that the integrality of these functions is equivalent to certain divisibility conditions on the number of ${\mathbb{F}}_p$ points on Jacobians of modular curves. Extending such expressions to arbitrary primes, we find trace functions for modules of cyclic groups of prime order with similar connections. Moreover, for cyclic groups, we give an explicit vertex operator algebra construction whose trace functions are given only in terms of weight two Eisenstein series.

18 pages

References in corpus (8)

• Vertex Algebras and Algebraic Curves
• Notes on the K3 Surface and the Mathieu group M_24
• Note on Twisted Elliptic Genus of K3 Surface
• Mathieu twining characters for K3
• Mathieu Moonshine in the elliptic genus of K3
• Eichler-Selberg Type Identities for Mixed Mock Modular Forms
• Super Vertex Algebras, Meromorphic Jacobi Forms and Umbral Moonshine
• The Umbral Moonshine Module for the Unique Unimodular Niemeier Root System

Cited by in corpus (1)

• On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras