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

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

2008-05-28

arxiv.org/abs/0805.4282v1

Mathematics

Number Theory

28 pages, 1 figure

Scientific paper

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.

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