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

2008/05/28 by Shuji Yamamoto, Shûji Yamamoto, Yamamoto, Shuji
Computer Science · Mathematics · #11M20 (Primary) 11R42 #11R80 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11M20 #msc:11R42 #msc:11R80

paper · pdf · doi:10.48550/arxiv.0805.4282

28 pages, 1 figure

arxiv created 2008/05/28 · openalex publication_date 2008/05/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=X₁... 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,χ) of L-functions.

Related

• Arithmetic of Hecke L-functions of quadratic extensions of totally real fields
• Ray class invariants over imaginary quadratic fields
• A proof of Sugawara's conjecture on Hasse-Weber ray class invariants
• Certain CM class fields with smaller generators
• La méthode de Shintani et ses variantes