Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Local monodromy of $ p$-divisible groups


Authors: Jeffrey D. Achter and Peter Norman
Journal: Trans. Amer. Math. Soc. 362 (2010), 985-1007
MSC (2000): Primary 14L05; Secondary 11S31
Published electronically: September 15, 2009
MathSciNet review: 2551513
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Abstract: A $ p$-divisible group over a field $ K$ admits a slope decomposition; associated to each slope $ \lambda$ is an integer $ m$ and a representation $ \mathrm{Gal}(K)\rightarrow\mathrm{GL}_m(D_\lambda)$, where $ D_\lambda$ is the $ \mathbb{Q}_p$-division algebra with Brauer invariant $ [\lambda]$. We call $ m$ the multiplicity of $ \lambda$ in the $ p$-divisible group. Let $ G_0$ be a $ p$-divisible group over a field $ k$. Suppose that $ \lambda$ is not a slope of $ G_0$, but that there exists a deformation of $ G$ in which $ \lambda$ appears with multiplicity one. Assume that $ \lambda\not= (s-1)/s$ for any natural number $ s>1$. We show that there exists a deformation $ G/R$ of $ G_0/k$ such that the representation $ \mathrm{Gal}(\mathrm{Frac} R) \rightarrow\mathrm{GL}_1(D_\lambda)$ has a large image.


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Additional Information

Jeffrey D. Achter
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email: j.achter@colostate.edu

Peter Norman
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
Email: norman@math.umass.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04818-1
Received by editor(s): May 30, 2006
Received by editor(s) in revised form: May 6, 2008
Published electronically: September 15, 2009
Additional Notes: The first author was partially supported by NSA grant H98230-08-1-0051.
Article copyright: © Copyright 2009 American Mathematical Society