On the second cohomology of the norm one group of a p-adic division algebra

TL;DR

This paper investigates the second cohomology group of the norm one group of a p-adic division algebra, providing explicit bounds and exact calculations under certain conditions, extending previous theoretical results.

Contribution

It offers explicit upper bounds for the second cohomology group and determines its structure in specific cases, advancing understanding of cohomology in p-adic division algebra groups.

Findings

Provides explicit upper bounds for H^2(SL_1(D), R/Z) for p ≥ 5.

Determines H^2(SL_1(D), R/Z) exactly when F is cyclotomic, p ≥ 19, and degree of D is not a p-power.

Extends previous results by Prasad and Raghunathan with concrete calculations.

Abstract

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 $p$.