Much ado about Mathieu

Eguchi, Ooguri and Tachikawa observed that the coefficients of the elliptic genus of type II string theory on K3 surfaces appear to be dimensions of representations of the largest Mathieu group. Subsequent work by several people established a candidate for the elliptic genus twisted by each element of M24M24. In this paper we prove that the resulting sequence of class functions are true characters of M24M24, proving the Eguchi–Ooguri–Tachikawa ‘Mathieu Moonshine’ conjecture. The integrality of multiplicities is proved using a small generalisation of Sturm's Theorem, while positivity involves a modification of a method of Hooley, for finding an effective bound on a family of Selberg–Kloosterman zeta functions at s=3/4s=3/4. We also prove the evenness property of the multiplicities, as conjectured by several authors, and use that to investigate the proposal of Gaberdiel–Hohenegger–Volpato that Mathieu Moonshine lifts to the Conway groups Co0Co0 and Co1Co1. We identify the role group cohomology plays in both Mathieu Moonshine and Monstrous Moonshine; in particular this gives a cohomological interpretation for the non-Fricke elements in Norton's Generalised Monstrous Moonshine conjecture, and gives a lower bound for H3(BCo1;C×)H3(BCo1;C×).