A relation between p-adic L-functions and the Tamagawa number conjecture for Hecke characters

Abstract.

We prove that the submodule in K-theory which gives the exact value $({\text{up to }}\mathbb{Z}_{(p)}^* )$ of the L-function by the Beilinson regulator map at non-critical values for Hecke characters of imaginary quadratic fields K with cl (K)  = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soulé regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsen’s conjecture, an upper bound for $\# H_{et}^2 (\mathcal{O}_K [1/S],\;V_p (m))$ in terms of the valuation of these p-adic L-functions, where V _p denotes the p-adic realization of a Hecke motive.