A pressure system from hyperbolic geometry to monstrous moonshine. Twenty parts, four acts, one destination.
The thesis. Modular forms begin as functions with severe symmetry. Their q-expansions turn those symmetries into arithmetic data. In the classical story that data counts divisors, detects elliptic curves, and builds L-functions. In moonshine the same mechanism goes one step further: the coefficients become dimensions and traces of representations of the Monster group. The series is built so each act earns the next constraint.
We move through four acts: geometry creates symmetry, symmetry creates scarcity, scarcity creates arithmetic signal, and one signal becomes representation theory. Three motifs recur at every level — the same value under many descriptions (modular invariance), boundary behaviour controls the whole object (cusps and q-expansions), and coefficients are witnesses (divisor sums, then Hecke eigenvalues, then elliptic-curve traces, then Monster characters).
Explore related topics in the Exceptional Atlas for more interactive algebra, or the information geometry series to see how these geometric ideas apply to probability distributions.
Each fact below is a constraint the series argues for. Read top-to-bottom and the four-act spine becomes a single arithmetic ladder — finite-dimensionality, two generators, a divisibility congruence, a unique invariant, an arithmetic signal, a representation-theoretic decomposition, a sporadic group, and finally a graded character.