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

TL;DR

This paper develops a combinatorial approach to describe Hecke L-functions of quadratic extensions over totally real fields, extending classical formulas and addressing conjectures in algebraic number theory.

Contribution

It introduces a combinatorial description of Shintani sets for certain characters, enabling explicit formulas for Hecke L-functions over quadratic extensions.

Findings

Provides a simple description of Hecke L-functions for specific characters

Extends class number formulas from imaginary to real quadratic extensions

Offers an effective approach towards Hecke's conjecture on class numbers

Abstract

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 $[F:\mathbb{Q}] = n \geq 3$, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of $F$ on $\mathbb{R}^n_+$, so-called $\textit{Shintani sets}$. 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.