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Paramodular abelian varieties of odd conductor. (English) Zbl 1285.11087

Trans. Am. Math. Soc. 366, No. 5, 2463-2516 (2014); corrigendum ibid. 372, No. 3, 2251-2254 (2019).
Summary: A precise and testable modularity conjecture for rational abelian surfaces \(A\) with trivial endomorphisms, \(\text{End}_{\mathbb Q}A=\mathbb Z\), is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen [Paramodular Cusp Forms, http://arxiv.org/pdf/0912.0049v1.pdf (2009)] 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.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

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