We prove that there are infinitely many real quadratic number fields $K$ with the property that for infinitely many orders $𝒪$ in $K$ and for the maximal order $R$ in $K$ the natural homomorphism $\varphi :W𝒪\to WR$ of Witt rings is surjective.

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $L\left(1,E/\mathbb{Q}({\mu }_{p^n},\sqrt[p^n]{\Delta })\right).$ 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 $K_1\left({\mathbb{Z}}_p\left[\left[\mathrm{Gal}(\mathbb{Q}({\mu }_{{\mathrm{p}}^{\infty }},\sqrt[{\mathrm{p}}^{\infty }]{\Delta })/\mathbb{Q})\right]\right]\right)$, 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.

In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum${\left(k\right)}_F$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum${\left(k\right)}_F$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric...