The modular curve X(N) is a fundamental object in number theory. As a Riemann surface, it is a quotient of the upper half plane by a subgroup of SL2(Z), but it also admits a moduli interpretation in terms of elliptic curves together with level structure. When p is a prime dividing N with high multiplicity, the standard model of X(N) over the integers has horrible singularities modulo p. We will reveal a new model for X(N) whose reduction modulo p is a kaleidoscopic configuration of interesting smooth curves modulo p, with only mild singularities (the model is "semistable"). This result is the tip of the iceberg of a story which unites the representation theory of p-adic groups with the geometry of varieties over finite fields.