Timeline for $p$-torsion in the Mordell-Weil group of Abelian varieties injecting in reduction
Current License: CC BY-SA 2.5
13 events
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Jun 24, 2010 at 19:39 | comment | added | Emerton | Dear Brian, Thanks for adding this clarification. | |
Jun 24, 2010 at 18:54 | comment | added | BCnrd | More specifically, I should have said that in the non-semistable case, the Zariski closure of $A[p]$ in $\mathcal{A}$ may be much smaller than $\mathcal{A}[p]$. But in the good reduction (or more generally, semistable) case these coincide, and it is the $p$-torsion in $\mathcal{A}$ that Emerton is denoting as $A[p]$ (under the quite standard and useful convention to let $A$ denote both the generic fiber as well as the N\'eron model, with the intended meaning conveyed through context). | |
Jun 24, 2010 at 18:51 | comment | added | BCnrd | To add to Emerton's clarification, if we let $\mathcal{A}$ denote the N\'eron model of $A$ over the local ring $\mathcal{O}_ {K, \mathfrak{p}}$ then the Zariski closure of $A[p]$ in $\mathcal{A}$ is always flat, and consequently quasi-finite (since the generic fiber is finite) even in cases with non-semistable reduction (for which $\mathcal{A}[p]$ is non-flat and has positive-dimensional special fiber). So it is really the finiteness that is the key place where Emerton invokes good reduction (since quasi-finite + proper implies finite, and $\mathcal{A}$ is proper in the good reduction case). | |
Jun 24, 2010 at 17:55 | history | edited | Emerton | CC BY-SA 2.5 |
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Jun 24, 2010 at 17:54 | comment | added | Emerton | To know that $A[p]$ is finite flat over the ring of integers. | |
Jun 24, 2010 at 17:41 | vote | accept | user6960 | ||
Jun 24, 2010 at 17:41 | comment | added | user6960 | Thanks for your response. My final question is: Where do we use that $A$ has good reduction at $\mathfrak{p}$? | |
Jun 24, 2010 at 5:06 | comment | added | BCnrd | As examples, consider an elliptic curve $E$ over $\mathbf{Q}$ with good reduction at 2 and full rational 2-torsion (impossible with odd primes in place of 2!): $E$ is defined by $y^2 = f(x)$ where $f$ is a split cubic over $\mathbf{Q}$. Then working over $\mathbf{Z}_2$, the $(\mathbf{Z}/(2))^2$ in the generic fiber over $\mathbf{Q}_2$ has schematic closure equal to the finite flat 2-torsion in the N\'eron model. Whether in the ordinary or supersingular reduction case, can then find a $\mathbf{Z}/(2)$ in the generic fiber whose closure has special fiber $\mu_2$ or $\alpha_2$ respectively. | |
Jun 23, 2010 at 21:07 | comment | added | Emerton | I added a more detailed explanation. | |
Jun 23, 2010 at 21:06 | history | edited | Emerton | CC BY-SA 2.5 |
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Jun 23, 2010 at 14:55 | comment | added | user6960 | I don't see why the extension is still a closed embedding and why the argument breaks down in the case $\mu_2$. | |
Jun 21, 2010 at 20:36 | comment | added | BCnrd | Matt's argument applies with $\mathbf{Z}/(p)$ replaced by a power, such as corresponding to basis of $A[p](K)$, so desired result follows. Sutble variant: is homomorphism $G' \rightarrow G$ between smooth affine gps over a dvr $R$ a closed immersion when it is on generic fibers? (Above is $G$ an abelian scheme and $G'$ a finite constant group.) In the important case that $G'$ has connected reductive fibers, it is OK away from exceptions for residue char. 2. This lies quite deep. See Cor. 1.3 in Prasad-Yu "Quasi-reductive groups" (not to be confused with pseudo-reductive groups...) | |
Jun 21, 2010 at 19:44 | history | answered | Emerton | CC BY-SA 2.5 |