1 results
Local–global principle for reduced norms over function fields of
$p$-adic curves
- Part of
-
- Journal:
- Compositio Mathematica / Volume 154 / Issue 2 / February 2018
- Published online by Cambridge University Press:
- 02 November 2017, pp. 410-458
- Print publication:
- February 2018
-
- Article
- Export citation
-
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 
$F$.
