Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set$${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series $\overline{f(T,{\theta }_{i,j})}$ are linearly independent over a certain...

For a typical example of a complete discrete valuation field $K$ of type II in the sense of [12], we determine the graded quotients ${gr}^i{K}_2\left(K\right)$ for all $i\>0$. In the Appendix, we describe the Milnor $K$-groups of a certain local ring by using differential modules, which are related to the theory of syntomic cohomology.

The purpose of this paper is to generalize, to certain commutative formal groups of dimension one and height greater than one defined over the ring of integers of a finite extension of ${\mathbf{Q}}_p$, some results on $p$-adic interpolation developed by Kubota, Leopoldt, Iwasawa, Mazur, Katz and others notably for the multiplicative group ${\widehat{\mathbf{G}}}_m$, and which they used to construct $p$-adic $L$-functions.

We construct $p$-adic $L$-functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.