An application of Birch-Tate formula to tame kernels of real quadratic number fields

2023/07/12 by Deng, Li-Tong, Li, Yong-Xiong
#11M06 11R42 (secondary) #11R70 (primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2307.05895

Abstract

Let F be a real quadratic number field with discriminant D and O_F the ring of integers in F. Let χ_F be the Dirichlet character associated to F/ℚ. Write L(χ_F,s) for the Dirichlet L-function of χ_F. By an induction argument for imprimitive Dirichlet L-values, we get several 2-divisibility results on L(χ_F,-1) when D has arbitrarily finitely many prime divisors. As an application, by making use of the Birch-Tate formula for F, we determine the 2-primary part for the second K group K₂O_F. We also give a new proof for an old theorem of Browkin and Schinzel.

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