Cyclic Algebras over $p$-adic curves

Details Cyclic Algebras over $p$-adic curves Cyclic Algebras over $p$-adic curves

2006-04-18

arxiv.org/abs/math/0604409v1

Mathematics

Algebraic Geometry

Scientific paper

In this paper we study division algebras over the function fields of curves over $\Q_p$. The first and main tool is to view these fields as function fields over nonsingular $S$ which are projective of relative dimension 1 over the $p$ adic ring $\Z_p$. A previous paper showed such division algebras had index bounded by $n^2$ assuming the exponent was $n$ and $n$ was prime to $p$. In this paper we consider algebras of degree (and hence exponent) $q \not= p$ and show these algebras are cyclic. We also find a geometric criterion for a Brauer class to have index $q$.

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