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Nov 11, 2013 at 6:28 comment added Marguax I should have mentioned that in 4.3 in Expose VII$_{\rm{A}}$ in SGA3 there is a discussion and construction of the relative Verscheibung homomorphism (and associated factorization of $[p]$ as $V \circ F$) for any flat commutative group scheme over an $\mathbf{F}_p$-scheme. This yields what I claimed at the level of formal groups above, when applied to the smooth Neron model over $R$.
Nov 11, 2013 at 6:19 comment added Marguax Since $A(K)$ is profinite separable and pro-$p$ near 0 with finite $p$-torsion, a small compact open subgroup $U$ in $A(K)$ has $U[p]=1$, so $U$ is a countable direct product of copies of $\mathbf{Z}_p$. It is a countably infinite product if $A(K)/(p)$ is infinite. Suppose $A(K)/(p)$ is finite, so $p\cdot A(K)$ is open in $A(K)$. In the formal group of the Neron model $N$ over $R$ with characteristic $p$, $[p]=V_N \circ F_N$. Thus, $[p]^{\ast}(X_j) \in R[[X_1^p,\dots,X_g^p]]$ for all $j$, so openness is impossible by measure-theoretic reasons. Thus, $U=\prod_{n=1}^{\infty}\mathbf{Z}_p$.
Nov 11, 2013 at 6:19 history edited Pete L. Clark CC BY-SA 3.0
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Nov 11, 2013 at 5:10 answer added Lubin timeline score: 5
Nov 11, 2013 at 4:18 history edited Pete L. Clark CC BY-SA 3.0
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Nov 11, 2013 at 3:05 history asked Pete L. Clark CC BY-SA 3.0