On the $p$-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Abstract

Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin $L$-functions attached to the irreducible characters of $G$ to those of corresponding complex Artin $L$-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the `$p$-part' of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(E))(r),\mathbb{Z}[G])$. If $r>1$ is even we obtain a similar result for Galois CM-extensions after restriction to `minus parts'.

arXiv comment32 pages, v2 contains considerable improvements (the vanishing of $\mu$ is no longer required); v3 minor changes and corrections, Remark 3.22 added; v4 further minor changes. To appear in Sel. Math. New Ser