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Abelian varieties having purely additive reduction. (English) Zbl 0557.14022
Let K be a field with a discrete valuation v with perfect residue field k. Let further A be an abelian variety of dimension g over K having purely additive reduction at v; this condition means that the connected component of the special fibre \(A_ 0\) of the Néron minimal model of A at v is a unipotent linear group over k. - For a prime number \(\ell\), let \(\ell^{b(\ell)}\) be the order of the \(\ell\)-primary part of A(K), with \(b(\ell)=\{0,1,2,...,\infty \}.\) The main theorem of this paper asserts that, with these hypotheses and notations, we have \(\sum (\ell - 1)b(\ell)\leq 2g,\) where the sum ranges over all prime numbers \(\ell \neq char(k)\). In particular A(K) has no \(\ell\)-torsion if \(\ell >2g+1.\)
The proof depends on monodromy arguments. Examples show that the inequality is best possible and that the restriction \(\ell \neq char(k)\) is essential. - It is a consequence of the main theorem that the number m of geometric components of \(A_ 0\) satisfies \(\sum (\ell - 1)ord_{\ell}(m)\leq 2g,\) where the sum is over the same \(\ell\). This improves a result of J. H. Silverman [Math. Ann. 264, 1-3 (1983; Zbl 0497.14016)]. It is not clear whether \(\ell =char(k)\) can be included in this sum. The paper concludes with some examples regarding the behaviour of points of order char(k) under the reduction map \(A(K)\to A_ 0(k).\)

MSC:
14K05 Algebraic theory of abelian varieties
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