Timeline for Can every finite abelian $p$-group with duality pairing be written as cokernel of a symmetric matrix over the $p$-adic integers?
Current License: CC BY-SA 3.0
19 events
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| May 3, 2020 at 14:04 | comment | added | LSpice | "The following paper" is Clancy, Kaplan, Leake, Payne, and Wood - On a Cohen–Lenstra heuristic for Jacobians of random graphs (now in v2). It's funny to call the pairing $\delta$ and then immediately write it as $\langle{}, {}\rangle$. :-) | |
| Jan 26, 2015 at 17:57 | comment | added | Pritam Majumder | @user74230: Yes, you are right, thanks. I have edited the question. | |
| Jan 26, 2015 at 17:46 | history | edited | Pritam Majumder | CC BY-SA 3.0 |
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| Jan 26, 2015 at 15:15 | comment | added | user74230 | @PritamMajumder: Your edit hasn't accurately expressed the question you have in mind, and you seem to have misunderstood Amritshanu Prasad's comment about your question literally as originally written. What you have not been explicitly saying but must have in mind is that the isomorphism has to respect the evident bilinear forms (valued in $\mathbf{Q}_p/\mathbf{Z}_p$) on both sides. Without that your question would have nothing to do with $\delta$; this is what others keep pointing out. | |
| Jan 26, 2015 at 11:45 | comment | added | Pritam Majumder | I have edited my question to explain the connection of the mentioned 'duality pairing'. | |
| Jan 26, 2015 at 11:44 | history | edited | Pritam Majumder | CC BY-SA 3.0 |
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| Jan 26, 2015 at 11:22 | comment | added | Neil Strickland | @AmritanshuPrasad's comment answers the question as asked, but the OP probably wanted some compatibility between $A$ and the given duality pairing. The OP should spell this out. | |
| Jan 26, 2015 at 10:37 | comment | added | Amritanshu Prasad | The structure theorem for finite abelian $p$-groups (or finite $\mathbf Z_p$-modules) says tha $G$ is a quotient of $\mathbf Z_p^n$ modulo quotient of a diagonal matrix of the form $A = \mathrm{diag}(p^{m_1},p^{m_2},\dotsc, p^{m_n})$. The existence of $\delta$ is not needed here. By the way I think it will also follow from the structure theorem that $\delta$ always exists. | |
| Jan 26, 2015 at 9:44 | history | edited | Dima Pasechnik | CC BY-SA 3.0 |
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| Jan 26, 2015 at 8:08 | comment | added | user74230 | Could whomever is down-voting this question indicate how they think about this without invoking facts concerning isotropicity of non-degenerate quadratic spaces of rank at least 3 over finite fields? | |
| Jan 26, 2015 at 8:02 | comment | added | user74230 | @abx: The structure of finite quadratic spaces and its relationship with $p$-adic quadratic spaces is reasonably elementary but not an totally trivial since it does involve input concerning the structure of non-degenerate quadratic spaces over finite fields. (It is not rocket science, but lots of stuff on MO is not rocket scence.) | |
| Jan 26, 2015 at 8:01 | answer | added | user74230 | timeline score: 2 | |
| Jan 26, 2015 at 7:49 | review | Close votes | |||
| Jan 26, 2015 at 23:03 | |||||
| Jan 26, 2015 at 7:41 | comment | added | Pritam Majumder | @abx: I did ask this question in MSE but unfortunately did not get any answer. Could you please elaborate your answer? | |
| Jan 26, 2015 at 7:32 | comment | added | abx | Then did you hear about the structure theorem for such groups? This is not a research level question, please use Mathstackexchange next time. | |
| Jan 26, 2015 at 7:11 | history | edited | Pritam Majumder | CC BY-SA 3.0 |
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| Jan 26, 2015 at 7:11 | comment | added | Pritam Majumder | Yes, finite and commutative. Edited, sorry for the confusion. | |
| Jan 26, 2015 at 6:58 | comment | added | abx | Is your group commutative? Finite? Finitely generated? | |
| Jan 26, 2015 at 6:16 | history | asked | Pritam Majumder | CC BY-SA 3.0 |