Local monodromy of $p$-divisible groups
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- by Jeffrey D. Achter and Peter Norman PDF
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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.References
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Additional Information
- Jeffrey D. Achter
- Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
- MR Author ID: 690384
- Email: j.achter@colostate.edu
- Peter Norman
- Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
- Email: norman@math.umass.edu
- 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.
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 985-1007
- MSC (2000): Primary 14L05; Secondary 11S31
- DOI: https://doi.org/10.1090/S0002-9947-09-04818-1
- MathSciNet review: 2551513