On the second cohomology of the norm one group of a p-adic division algebra
| Autor: | Ershov, Mikhail, Weigel, Thomas |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb Q_p$. Let $D$ be a finite-dimensional central division algebra over $F$ and let $SL_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and Raghunathan proved that $H^2(SL_1(D),\mathbb R/\mathbb Z)$ is a cyclic $p$-group whose order is bounded from below by the number of $p$-power roots of unity in $F$, unless $D$ is a quaternion algebra over $\mathbb Q_2$. In this paper we give an explicit upper bound for the order of $H^2(SL_1(D),\mathbb R/\mathbb Z)$ for $p\geq 5$ and determine $H^2(SL_1(D),\mathbb R/\mathbb Z)$ precisely when $F$ is cyclotomic, $p\geq 19$ and the degree of $D$ is not a power of $p$. Comment: 67 pages. This is a paper by the first author with an appendix by both authors. To appear in the Michigan Mathematical Journal |
| Databáze: | arXiv |
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