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

## TL;DR

This paper explores the connections between the $p$-adic Beilinson conjecture, the equivariant Iwasawa main conjecture, and the equivariant Tamagawa number conjecture, establishing implications for Galois extensions of totally real fields.

## Contribution

It demonstrates that these conjectures imply the $p$-part of the equivariant Tamagawa number conjecture for certain Galois extensions, extending results to CM-extensions for even $r$.

## Key findings

- Implication of the $p$-adic Beilinson conjecture on Tamagawa number conjecture.
- Reduction of conjectures to the $p$-part for Galois extensions.
- Extension of results to CM-extensions with even $r$.

## 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'.