Timeline for Conjugacy for $p$-adic matrices of finite order
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 24, 2023 at 17:46 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator`
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| Jan 24, 2023 at 17:02 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed, cf. https://meta.mathoverflow.net/q/5301/70594
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| Aug 6, 2011 at 9:19 | comment | added | Junkie | However, the general Jordan form for $\Phi_{p^3}$ is $p$ copies of the $\Phi_{p^2}$ matrix glued by 1's, while the direct sum from $\Phi_{p^2}^p$ does not have these extra 1's. I think it is true that the general Jordan forms of companion matrices of factors of cyclotomic polynomials, when taken modulo $p$ are enough distinct that the mod $p$ decomposition determines that in $Z_p$. | |
| Aug 6, 2011 at 4:00 | comment | added | Junkie | You are right, Qiaochu Yuan, the best answer above noted in essence that $\Phi_{p^3}\equiv\Phi_{p^2}^p$ modulo $p$, and I did not think this to be possible, but I think I should have suspected this, for how $p$th powers work modulo $p$. | |
| Aug 5, 2011 at 14:27 | comment | added | Qiaochu Yuan | @Junkie: really? I would have expected problems if $p | m$ as these results suggest (by analogy with Hensel's lemma, for example). | |
| Aug 5, 2011 at 12:24 | comment | added | Junkie | I had the opposite thought. The forcing of $A,B$ to have finite order in $GL_n(Z_p)$ is so strong, that it eliminates any mischief on the periphery of your above theorems. Yet I also don't have a proof, or counterexample. The essential line of thought, which I made was to conjugate to blocks of a general Jordan form (companion form of a cyclotomic polynomial), and the finite order condition restricts the glueing elements to be zero. Then if this simplification is ok, it seems a tricky question about whether two cyclotomic polynomials can be congruent mod an odd prime. | |
| Aug 5, 2011 at 11:57 | comment | added | user91132 | The case $m = p$ and $n = p-1$ is particularly interesting. | |
| Aug 5, 2011 at 11:43 | history | answered | Gjergji Zaimi | CC BY-SA 3.0 |