Abstract
We introduce a ray class invariant \({X(\mathfrak{C})}\) for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula \({X(\mathfrak{C})=X_1(\mathfrak{C})\cdots X_n(\mathfrak{C})}\) where each \({X_i(\mathfrak{C})}\) 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 \({X_i(\mathfrak{C})}\) when the signature of \({\mathfrak{C}}\) at a real place is changed. This last result is also interpreted in terms of the derivatives L′(0, χ) of the L-functions and certain Stark units.
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Yamamoto, S. On Shintani’s ray class invariant for totally real number fields. Math. Ann. 346, 449–476 (2010). https://doi.org/10.1007/s00208-009-0405-x
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DOI: https://doi.org/10.1007/s00208-009-0405-x