Modular Forms

From hyperbolic geometry to monstrous moonshine, in fourteen parts and four acts.

Modular forms are functions with a great deal of symmetry, and their q-expansions turn that symmetry into arithmetic. In the classical story the coefficients count divisors, detect elliptic curves and build L-functions. In moonshine they become dimensions and traces of representations of the Monster group.

The first act builds the symmetry from the geometry of the upper half-plane. The second shows how few functions survive it. The third reads their coefficients as arithmetic, and the fourth reads one particular set of coefficients as Monster characters. Throughout, the coefficients carry the information: divisor sums, then Hecke eigenvalues, then point counts on elliptic curves, then characters.

The Leech lattice, from which the moonshine module is built, is covered with E₈ in the Exceptional Atlas.