Timeline for What is the structure of the group of rational points of an abelian variety over a Laurent series field?
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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 |