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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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References:

[1] Z. Borevich and I. Shafarevič , Number Theory , Academic Press, New York (1966). · Zbl 0145.04902
[2] E.J. Ditters , Higher Hasse-Witt matrices, Journées de Géometrie Algébrique de Rennes 1 (1978), Astérisque, Soc. Math. de Fr., 63 (1979) 67-71. · Zbl 0426.14021
[3] A. Grothendieck , Technique de descente et theoremes d’existence en géometrie algébrique , Séminaire Bourbaki 190 (1959). · Zbl 0229.14007
[4] M. Hazewinkel , Formal Groups and Applications . Academic Press: New York (1978). · Zbl 0454.14020
[5] J. Lubin , The canonicity of a cyclic subgroup of an elliptic curve , Journées de Géometrie Algébrique de Rennes 1 (1978), Astérique, Soc. Math. de Fr. 63 (1979) 165-167. · Zbl 0426.14020
[6] J. Lubin , Canonical subgroups of formal groups . Trans. Am. Math. Soc. 251 (1979) 103-127. · Zbl 0431.14014 · doi:10.2307/1998685
[7] J. Lubin and J. Tate , Formal moduli for one parameter formal Lie groups , Bull. Soc. Math. de Fr. 85 (1967) 49-60. · Zbl 0156.04105 · doi:10.24033/bsmf.1633
[8] B. Mazur , Local flat duality , Amer. J. of Math. 92 (1970) 201-223. · Zbl 0199.24501 · doi:10.2307/2373327
[9] B. Mazur and L. Roberts , Local Euler characteristic. Inventiones Math. 9 (1970) 201-234. · Zbl 0191.19202 · doi:10.1007/BF01404325
[10] L. Roberts , The flat cohomology of group schemes, Ph.D. Thesis , Harvard Univ. (1968).
[11] L. Roberts , The flat cohomology of group schemes of rank p , Am. J. Math. 95 (1973) 688-702. · Zbl 0281.14020 · doi:10.2307/2373735
[12] J. Tate , p-divisible groups , Proc. of a Conf. on Local Fields, T.A. Springer, Springer-Verlag, Berlin and New York (1979) pp. 158-183. · Zbl 0157.27601
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