关于p进除法代数的范一群的第二上同调

Pub Date : 2022-06-26 DOI:10.1307/mmj/20217210
M. Ershov, T. Weigel
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引用次数: 2

摘要

设F是p进域,即Qp的有限扩展。设D是F上的有限维除法代数,设SL1(D)是D上约简范数1的元素群。Prasad和Raghunathan证明了H(SL1(D), R/Z)是一个循环p群,其阶由F上的单位p次根的个数从下限定,除非D是Q2上的四元数代数。本文给出了p≥5时H(SL1(D), R/Z)阶的显式上界,并精确地确定了当F是分环的,p≥19,D的阶不是p的幂时H(SL1(D), R/Z)的阶。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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On the Second Cohomology of the Norm One Group of a p-Adic Division Algebra
Let F be a p-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H(SL1(D), R/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 Q2. In this paper we give an explicit upper bound for the order of H(SL1(D), R/Z) for p ≥ 5, and determine H(SL1(D), R/Z) precisely when F is cyclotomic, p ≥ 19 and the degree of D is not a power of p.
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