On Shintani's ray class invariant for totally real number fields

TL;DR

This paper introduces a new ray class invariant for totally real fields, proves its factorization into local components, and explores its independence from certain choices and its relation to derivatives of L-functions.

Contribution

It extends Shintani's work by defining a ray class invariant for totally real fields and establishing its factorization and invariance properties.

Findings

Proved the factorization of the invariant into local factors.

Showed the independence of the factorization from cone decomposition choices.

Linked the behavior of the invariant to the derivatives of L-functions.

Abstract

We introduce a ray class invariant $X(C)$ for a totally real field, following Shintani's work in the real quadratic case. We prove a factorization formula $X=X_1... X_n$ where each $X_i=X_i(C)$ corresponds to a real place. Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices. Finally, we describe the behavior of $X_i(C)$ when the signature of $C$ at a real place is changed. This last result is also interpreted into an interesting behavior of the derivative $L'(0,\chi)$ of $L$-functions.