A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface S and corresponding invariants of the Hilbert scheme of n points on S. This relationship is encoded in certain infinite product q-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these q-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the q-series' coefficients.
In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O'Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional C[G]-modules where the McKay-Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on Γ 0 (ord(g)). As there are only finitely many normalized Hauptmoduln, groups whose McKay-Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level ord(g) as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include (Z/5Z) 5 and (Z/7Z) 4 , which are not subgroups of the Monster.
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