Timeline for Diferent abelian varieties over local field with the same p-adic representation?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2013 at 9:05 | vote | accept | TOM | ||
| Jan 15, 2013 at 17:20 | comment | added | David Loeffler | @pranavk: Sorry, didn't see your comment -- you said considerably more than I did in less space! | |
| Jan 15, 2013 at 17:18 | answer | added | David Loeffler | timeline score: 10 | |
| Jan 15, 2013 at 17:16 | comment | added | user30379 | Addendum to previous comment: one can make a variant in the good ordinary reduction case, except that the case of decomposable $V_p(E)$ (i.e., Serre-Tate canonical lifts) needs to be excluded. In the multiplicative reduction case there is an affirmative result near the end of Serre's book "Abelian $\ell$-adic representations" (using calculations with Tate curves). The omitted details in my comments are good exercises, so I'll leave it at that. | |
| Jan 15, 2013 at 17:05 | comment | added | user30379 | Yes. Consider elliptic curves $E$ over $K=\mathbf{Q}_p$ with good ss reduction, so $V_p(E)$ is irreducible and crystalline with Hodge-Tate weights $0$ and $1$. By a calculation in $p$-adic Hodge theory (for 2-dimensional representations and distinct Hodge-Tate weights), $V_p(E)$ is determined up to abstract isomorphism by the char. polynomial of the linear (!) Frobenius on the Dieudonn\'e module of the reduction $E_0$. But geometric isogeny classes mod isom. are countable and the deformation ring of $E_0$ has uncountably many $\mathbf{Q}_p$-points, so lots aren't $\overline{K}$-isogenous. | |
| Jan 15, 2013 at 17:00 | comment | added | Damian Rössler | See mathoverflow.net/questions/53014/… | |
| Jan 15, 2013 at 16:01 | history | edited | Leo Alonso |
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| Jan 15, 2013 at 16:01 | history | edited | Leo Alonso |
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| Jan 15, 2013 at 16:00 | history | edited | Leo Alonso |
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| Jan 15, 2013 at 15:31 | history | asked | TOM | CC BY-SA 3.0 |