Paramodular abelian varieties of odd conductor

Brumer, Armand; Kramer, Kenneth

doi:10.1090/S0002-9947-2013-05909-0

Paramodular abelian varieties of odd conductor
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by Armand Brumer and Kenneth Kramer PDF

Trans. Amer. Math. Soc. 366 (2014), 2463-2516 Request permission

Corrigendum: Trans. Amer. Math. Soc. 372 (2019), 2251-2254.

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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