Department of Mathematics

Algebra

  •  Sumit Chandra Mishra, Emory University
  •  Local-global principle for norms over semi-global fields
  •  10/23/2019
  •  3:00 PM - 3:50 PM
  •  C304 Wells Hall

Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $F$ be a function field in one variable over $K$ and $\mathscr{X}$ a regular proper model of $F$ with reduced special fibre $X$ a union of regular curves with normal crossings. Suppose that the graph associated to $\mathscr{X}$ is a tree (e.g. $F = K(t)$). Let $L/F$ be a Galois extension of degree $n$ with Galois group $G$ and $n$ coprime to char$(\kappa)$. Suppose that $\kappa$ is algebraically closed field or a finite field containing a primitive $n^{\rm th}$ root of unity. Then we show that an element in $F^*$ is a norm from the extension $L/F$ if it is a norm from the extensions $L\otimes_F F_\nu/F_\nu$ for all discrete valuations $\nu$ of $F$.

 

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Michigan State University
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