Cyclotomic fields are generated by cyclotomic Hecke L-values of totally real fields

Abstract

For an unramified odd prime p in a totally real field, we prove that a single Hecke L-value of the field with cyclotomic twist of a sufficiently large p-power conductor, generates the p-power roots of unity. The proof involves technical but crucial ingredients, namely estimations both on the size of an exponential sum over the unit group and on the number of lattice points in a cone with a given congruence.