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Keywords:
algebraic $K$-theory; Dedekind zeta function; Artin $L$-function; Beilinson regulator; generalized index; Lichtenbaum conjecture
algebraic $K$-theory; Dedekind zeta function; Artin $L$-function; Beilinson regulator; generalized index; Lichtenbaum conjecture
Summary:
Let $F/k$ be a finite abelian extension of number fields with $k$ imaginary quadratic. Let $O_F$ be the ring of integers of $F$ and $n\geq 2$ a rational integer. We construct a submodule in the higher odd-degree algebraic $K$-groups of $O_F$ using corresponding Gross's special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher ``twisted'' class number of $F$, which is the cardinal of the finite algebraic $K$-group $K_{2n-2}(O_F)$.
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