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Local–global principle for reduced norms over function fields of $p$-adic curves

Part of: Algebraic number theory: local and $p$-adic fields General nonassociative rings Arithmetic algebraic geometry Forms and linear algebraic groups

Published online by Cambridge University Press:  02 November 2017

R. Parimala ,
R. Preeti  and
V. Suresh
R. Parimala
Affiliation:
Department of Mathematics & Computer Science, Emory University, 400 Dowman Drive NE, Atlanta, GA 30322, USA email [email protected]
R. Preeti
Affiliation:
Department of Mathematics, Indian Institute of Technology (Bombay), Powai, Mumbai-400076, India email [email protected]
V. Suresh
Affiliation:
Department of Mathematics & Computer Science, Emory University, 400 Dowman Drive NE, Atlanta, GA 30322, USA email [email protected]
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Abstract

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 $F$.

Keywords

division algebrasreduced normsfunction field of p-adic curvesGalois cohomology

MSC classification

Primary: 11E72: Galois cohomology of linear algebraic groups 11G99: None of the above, but in this section 11S25: Galois cohomology 17A35: Division algebras
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 2 , February 2018 , pp. 410 - 458
Copyright
© The Authors 2017 

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