Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for $[F:Q]=n\ge 3$, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on $R_{+}^n$, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.