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Purely additive reduction of abelian varieties with torsion. (English) Zbl 1491.11061

Summary: Let \(\mathcal{O}_K\) be a discrete valuation ring with fraction field \(K\) of characteristic 0 and algebraically closed residue field \(k\) of characteristic \(p > 0\). Let \(A / K\) be an abelian variety of dimension \(g\) with a \(K\)-rational point of order \(p\). In this article, we are interested in the reduction properties that \(A / K\) can have. After discussing the general case, we specialize to \(g = 1\), and we study the possible Kodaira types that can occur.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
14H25 Arithmetic ground fields for curves
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