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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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