Non-abelian congruences between $L$-values of elliptic curves

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $ L\bigl (1, E/\mathbb{Q}( \mu _{p^n},\@root p^n \of {\Delta } )\bigr ) . $ More precisely, we show that they are true modulo $p^{n+1}$, rather than modulo $p^{2n}$. Whilst not quite enough to establish that there is a non-abelian $L$-function living in $K_1\bigl ( \mathbb{Z}_p[[ \rm {Gal} ( \mathbb{Q}( \mu _{p^\infty },\!\!\@root p^\infty \of {\Delta } )/\mathbb{Q}) ]] \bigr )$, they do provide strong evidence towards the existence of such an analytic object. For example, if $n=1$ these verify the numerical congruences found by Tim and Vladimir Dokchitser.