Cyclotomic fields are generated by cyclotomic Hecke {\it L}-values of totally real fields, II

TL;DR

This paper extends previous work on generating cyclotomic fields using Hecke L-values to include non-trivial characters over solvable totally real fields, employing advanced number theoretic tools.

Contribution

It generalizes the generation of cyclotomic fields by critical L-values to non-trivial Hecke characters over solvable totally real fields, using new analytic and algebraic techniques.

Findings

Successfully extends generation results to non-trivial characters

Utilizes global class field theory and duality principles

Establishes non-vanishing of certain exponential sums

Abstract

Jun-Lee-Sun posed the question of whether the cyclotomic Hecke field can be generated by a single critical $L$-value of a cyclotomic Hecke character over a totally real field. They provided an answer to this question in the case where the tame Hecke character is trivial. In this paper, we extend their work to address the case of non-trivial Hecke characters over solvable totally real number fields. Our approach builds upon the primary estimation obtained by Jun-Lee-Sun, supplemented with new inputs, including global class field theory, duality principles, the analytic behavior of partial Hecke $L$-functions, and the non-vanishing of twisted Gauss sums and Hyper Kloosterman sums.