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Paramodular abelian varieties of odd conductor


Authors: Armand Brumer and Kenneth Kramer
Journal: Trans. Amer. Math. Soc. 366 (2014), 2463-2516
MSC (2010): Primary 11G10; Secondary 14K15, 11F46
Published electronically: November 21, 2013
MathSciNet review: 3165645
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Abstract: A precise and testable modularity conjecture for rational abelian surfaces $ A$ with trivial endomorphisms, $ \mathrm {End}_{\mathbb{Q}} A = \mathbb{Z}$, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on $ \ell $-division fields of semistable abelian varieties, mainly when $ A[\ell ]$ is reducible, by considering extension problems for group schemes of small rank.


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Additional Information

Armand Brumer
Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458
Email: brumer@fordham.edu

Kenneth Kramer
Affiliation: Department of Mathematics, Queens College, Flushing, New York 11367 – and – The Graduate Center CUNY, New York, New York 10016
Email: kkramer@gc.cuny.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05909-0
Keywords: Abelian variety, finite flat group scheme, polarization, division field, paramodular group
Received by editor(s): May 24, 2010
Received by editor(s) in revised form: July 4, 2012
Published electronically: November 21, 2013
Additional Notes: The research of the second author was partially supported by NSF grant DMS 0739346
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.