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#### Seminar der Forschergruppe 'Symmetrie, Geometrie und Arithmetik'

„K-theoretic methods in the representation theory of p-adic analytic groups“
M.Sc. Tamás Csige, Humboldt-Universität zu Berlin

Let $p$ be a prime number such that $p\geq 5$. Let $G=H \times Z$, where $H$ is a torsion free compact $p$-adic analytic group such that its Lie algebra is split semisimple over $\mathbb{Q}_p$ and $Z \cong \mathbb{Z}^n_p$, where $n \ge 0$. Let $M$ be a finitely generated torsion module over the Iwasawa algebra $\Lambda_G$ of $G$ such that it has no non-zero pseudo-null submodules. Let $q(M)$ denote the image of $M$ in the quotient category mod-$\Lambda_G/\mathcal{C}^1_{\Lambda_G}$ via the quotient functor $q$, where $\mathcal{C}^1_{\Lambda_G}$ denotes the Serre-subcategory of pseudo-null $\Lambda_G$-modules of the category of finitely generated $\Lambda_G$-modules, denoted by mod-$\Lambda_G$. Then $q(M)$ is completely faithful if and only if $M$ is $\Lambda_Z$-torsion free. This result is the generalization of the main theorem of \citep{Ar1}. We denote by $\mathfrak{N}_H(G)$, the category of finitely generated $\Lambda_G$-modules that are also finitely generated as $\Lambda_H$-modules. Let $M$, $N \in \mathfrak{N}_H(G)$ such that they have no non-zero pseudo-null $\Lambda_G$-submodules and let $q(M)$ be completely faithful. If $[M] = [N]$ in $K_0(\mathfrak{N}_H(G))$ then $q(N)$ is also completely faithful. \newline \noindent For the second part of the talk: Let $G$ be an arbitrary compact $p$-adic analytic group with no element of order $p$. Choose an open normal uniform pro-$p$ subgroup $H$ of $G$. Let $K$ be a finite extension of $\mathbb{Q}_p$ such that it contains all the $n$-th roots of unity, where $n:=|G/H|$. Define $K[[G]] := K \otimes_{\mathbb{Z}_p} \Lambda_G$ (it is called the algebra of continuous distributions). Then $K_0(K[[G]]) \cong \mathbb{Z}^c$, where $c$ is the number of conjugacy classes of $G/H$ of order relative prime to $p$. Moreover, if $r \in p^{\mathbb{Q}}$ such that $1/p < r<1$ then $K_0(\drho)$ is isomorphic to $\mathbb{Z}^c$, where $\drho$ is the algebra of bounded distributions of $G$ and $c$ is the same as above. I also mention some corollaries of this theorem. \begin{thebibliography}{98} \bibitem{Ar1} Ardakov K.: Centres of Skewfields and completely faithful Iwasawa modules. \emph{ J. Inst. Math. Jussieu} \textbf{7} (2008). \bibitem{C} Coates J., Fukaya T., Kato K., Sujatha R., Venjakob O.: The GL2 main conjecture for elliptic curves without complex multiplication, \emph{Publ. Math. IHES} \textbf{101} (2005), 163-208. \end{thebibliography}

Freitag, den 9. Dezember 2016 um 13:30 Uhr, in INF205, SR C Freitag, den 9. Dezember 2016 at 13:30, in INF205, SR C

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. Dr. Otmar Venjakob