For the $(d+1)$-dimensional Lie group $G=\mathbb{Z}_{p}^{\times }\ltimes \mathbb{Z}_{p}^{\oplus d}$, we determine through the use of $p$-power congruences a necessary and sufficient set of conditions whereby a collection of abelian $L$-functions arises from an element in $K_{1}(\mathbb{Z}_{p}\unicode[STIX]{x27E6}G\unicode[STIX]{x27E7})$. If $E$ is a semistable elliptic curve over $\mathbb{Q}$, these abelian $L$-functions already exist; therefore, one can obtain many new families of higher order $p$-adic congruences. The first layer congruences are then verified computationally in a variety of cases.

If is an integral model of a smooth curve X over a global field k, there is a localization sequence comparing the K-theory of and X. We show that K1 () injects into K1(X) rationally, by showing that the previous boundary map in the localization sequence is rationally a surjection, for X of “GL2 type” and k of positive characteristic not 2. Examples are given to show that the relative G1 term can have large rank. Examples of such curves include non-isotrivial elliptic curves, Drinfeld modular curves, and the moduli of -elliptic sheaves of rank 2.

We establish various properties of the definition of cohomology of topological groups given by Grothendieck, Artin and Verdier in SGA4, including a Hochschild–Serre spectral sequence and a continuity theorem for compact groups. We use these properties to compute the cohomology of the Weil group of a totally imaginary field, and of the Weil-étale topology of a number ring recently introduced by Lichtenbaum (both with integer coefficients).