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Selmer group estimates arising form the existence of canonical subgroups. (English) Zbl 0751.14030
Author’s abstract: Generalizing the work of Lubin in the one-dimensional case, conditions are found for the existence of canonical subgroups of finite height commutative formal groups of arbitrary dimension over local rings of mixed characteristic \((0,p)\). These are \(p\)-torsion subgroups which are optimally close to being kernels of Frobenius homomorphisms. The \(\mathbb{F}_ p\) ranks of the first flat cohomology groups of these canonical subgroups are found. These results are applied to the estimation of the \(\mathbb{F}_ p\) rank of the Selmer group of an Abelian variety over a global number field of characteristic zero, and the \(\lim\sup\) of these ranks as the Abelian variety varies in an isogeny class.
MSC:
14L05 Formal groups, \(p\)-divisible groups
14K05 Algebraic theory of abelian varieties
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