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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaabw % hacaqGWbGaaeiiaiaabshacaqGVbGaaeiiamrr1ngBPrwtHrhAYaqe % guuDJXwAKbstHrhAGq1DVbacfaGae8hjHO1aa0baaSqaaiaacIcaca % WGWbGaaiykaaqaaiaacQcaaaGccaGGPaaaaa!4A5B! $$({\text{up to }}\mathbb{Z}_{(p)}^* )$$ 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 Soule 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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4iaiaadI % eadaqhaaWcbaGaamyzaiaadshaaeaacaaIYaaaaOGaaiikamrr1ngB % PrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8NdX-0aaSbaaS % qaaiaadUeaaeqaaOGaai4waiaaigdacaGGVaGaam4uaiaac2facaGG % SaGaaGjbVlaadAfadaWgaaWcbaGaamiCaaqabaGccaGGOaGaamyBai % aacMcacaGGPaaaaa!5288! $$\# H_{et}^2 (\mathcal{O}_K [1/S],\;V_p (m))$$ in terms of the valuation of these p-adic L-functions, where V p denotes the p-adic realization of a Hecke motive.