Elliptic unit

Elliptic units are algebraic units constructed by evaluating suitably normalized elliptic functions at CM torsion points and taking products or norms into ray class fields. Their distribution relations make them norm compatible in CM division towers. They generate an Euler system of elliptic units and their local Coleman power series for a CM formal group have logarithmic derivatives encoding critical Hecke L-function values.

Elliptic-unit distribution gives norm-compatible units indexed by auxiliary squarefree ray moduli, with norm factor $1-{\mathrm{Frob}}_{\mathfrak{q}}^{-1}$ at a newly added prime. Kolyvagin derivative operators turn these units into classes with controlled valuations, and suitable auxiliary primes chosen using Chebotarev density theorem bound class-group characteristic ideals. The construction supplies the difficult divisibility in the CM Iwasawa main conjecture.