The Hecke field generation is one of the important problems in number theory as important arithmetic consequences arise. For example, Sun proved the generation of Hecke fields of newforms by modular L-values twisted by cyclotomic characters and applied it to the growth of Hecke fields in a Hida family. Recently, Jun-Lee-Sun proved that the cyclotomic fields are generated by L-values of cyclotomic characters over totally real fields by utilizing the non-singular cone decomposition of the unit cone of totally real fields.
We prove a result toward the generation of cyclotomic Hecke fields by L-values of cyclotomic Hecke characters over totally real fields, which is a generalization of Jun-Lee-Sun. Note that this result is stronger than the non-vanishing of L-values. Main ingredients of the proof are multi-dimensional Kloosterman sums, global class field theory, and Jun-Lee-Sun's estimation on the twisted average of L-values.
This work is joint with Hae-Sang Sun.