This volume contains the proceedings of the Moonshine workshop held at the Centre de Recherches Mathématiques (CRM) in Montréal. A glance at the contents will reveal that the connection of some papers to Moonshine is not immediate; however, Moonshine has proved to be a very fertile area, and it does not stretch the imagination to believe that many more threads will be drawn together before we understand what is really going on.

In this volume, all the classical Moonshine themes are presented, namely the Monster simple group and other finite groups, automorphic functions and forms and related congruence groups, and vertex algebras and their representations. These topics appear in either a pure form or in a blend of algebraic geometry dealing with algebraic surfaces, Picard-Fuchs equations, and hypergeometric functions.

Contents

• A. Baker and H. Tamanoi -- Invariants for finite dimensional groups in vertex operator algebras associated to basic representations of affine algebras
• C. Dong and G. Mason -- Transformation laws for theta functions
• C. F. Doran -- Algebro-geometric isomonodromic deformations linking Hauptmoduls: Variation of the mirror map
• G. Glauberman and S. P. Norton -- On McKay's connection between the affine $E_8$ diagram and the monster
• K. Harada and M. L. Lang -- Sylow 2-subgroups of simple groups
• W. L. Hoyt and C. F. Schwartz -- Yoshida surfaces with Picard number $\rho \geq 17$
• M. Kaneko and N. Todaka -- Hypergeometric modular forms and supersingular elliptic curves
• C. H. Lam -- Fusion rules for ternary and $\mathbb{Z}_2 \times \mathbb{Z}_2$ code vertex operator algebras
• H. Li -- The regular representations and the $A_{n}(V)$-algebras
• J. McKay -- Linear dependencies among completely replicable functions
• J. McKay and A. Sebbar -- Arithmetic semistable elliptic surfaces
• M. Miyamoto -- Modular invariance of trace functions on VOAs in many variables
• N. Narumiya and H. Shiga -- The mirror map for a family of $K$3 surfaces induced from the simplest 3-dimensional reflexive polytope
• S. Norton -- From moonshine to the monster
• Y. Ohyama -- Hypergeometric functions and non-associative algebras
• K. Saito -- Extended affine root systems. V. Elliptic eta-products and their Dirichlet series
• C. S. Simons -- Deflating infinite Coxeter groups to finite groups
• M. P. Tuite -- Genus two meromorphic conformal field theory
• H. Verrill -- Picard-Fuchs equations of some families of elliptic curves