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

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 $L^{\prime }(0,\chi )$ of $L$-functions.

28 pages, 1 figure