in 1844, kummer showed that cyclotomic integer rings can fail to be principal ideal domains. in 1976 and 1984, ribet and mazur-wiles used galois representations attached to modular forms to partially describe class groups that measure the extent of this failure. the exact structure of these class groups remains a mystery to this day. i will explain how to attach ideal classes to geodesics in the complex upper half-plane. a conjecture of mine states these two constructions are inverse to each other in an appropriate sense. i hope to motivate a broader philosophy, developed jointly with takako fukaya and kazuya kato, that certain arithmetic objects attached to galois representations of global fields can be described using higher-dimensional modular symbols.