Local–global principle for reduced norms over function fields of -adic curves
2017 ◽
Vol 154
(2)
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pp. 410-458
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Keyword(s):
Let $K$ be a (non-archimedean) local field and let $F$ be the function field of a curve over $K$. Let $D$ be a central simple algebra over $F$ of period $n$ and $\unicode[STIX]{x1D706}\in F^{\ast }$. We show that if $n$ is coprime to the characteristic of the residue field of $K$ and $D\cdot (\unicode[STIX]{x1D706})=0$ in $H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$, then $\unicode[STIX]{x1D706}$ is a reduced norm from $D$. This leads to a Hasse principle for the group $\operatorname{SL}_{1}(D)$, namely, an element $\unicode[STIX]{x1D706}\in F^{\ast }$ is a reduced norm from $D$ if and only if it is a reduced norm locally at all discrete valuations of $F$.
1966 ◽
Vol 27
(2)
◽
pp. 625-642
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Keyword(s):
2017 ◽
Vol 13
(04)
◽
pp. 853-884
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Keyword(s):
2018 ◽
Vol 2018
(745)
◽
pp. 41-58
2018 ◽
Vol 17
(12)
◽
pp. 1850240
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Keyword(s):
1994 ◽
Vol 37
(3)
◽
pp. 445-454
Keyword(s):
2015 ◽
Vol 14
(04)
◽
pp. 1550048
◽
2018 ◽
Vol 222
(9)
◽
pp. 2773-2783
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