Timeline for Is there a good notion of kernels of quadratic forms on abelian groups?
Current License: CC BY-SA 4.0
14 events
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| Mar 10, 2023 at 22:40 | comment | added | Dave Benson | Can I recommend Ed Brown's paper, "Generalizations of the Kervaire invariant," especially theorem 1.20 in that paper. I learned a great deal about quadratic forms and the prime two from this paper, especially the fact that there is a mod 8 invariant and a Gauß sum that computes it. | |
| Feb 27, 2023 at 7:53 | vote | accept | Bipolar Minds | ||
| Feb 26, 2023 at 10:08 | answer | added | Uriya First | timeline score: 2 | |
| Feb 20, 2023 at 14:29 | comment | added | Theo Johnson-Freyd | @UriyaFirst and Ycor: Perhaps leave this as an answer rather than a comment? | |
| Feb 19, 2023 at 17:51 | comment | added | Bipolar Minds | Ah, so its the (usual) kernel of $q|_{\operatorname{rad(b)}}$ as $q$ is a homomorphism on $\operatorname{rad(b)}$ | |
| Feb 19, 2023 at 13:09 | comment | added | Uriya First | In the theory of quadratic forms over a field of characteristic 2, the radical of a quadratic form $q$ is sometimes defined as $R(q)=\{x\in \mathrm{rad}(b):q(x)=0\}$, where $b$ is the polar form of $q$. This should work in your situation as well: $R(q)$ is a subgroup of $G$ and $q$ factors via a $\mathbb{Q}/\mathbb{Z}$-quadratic map on $G/R(q)$. (This should coincide with @YCor's suggestion, but is perhaps conceptually clearer.) | |
| Feb 19, 2023 at 12:36 | comment | added | Bipolar Minds | @YCor Thx, that makes sense. Using the above notation the factor map $\bar{q}$ should be again a quadratic form since the quotient map $\pi$ is an epimorphism, right? Still I would like to have a more 'compact' description of this group, but I first need to make up my mind about what I mean by that.. | |
| Feb 19, 2023 at 10:59 | comment | added | YCor | A "kernel" can be defined for an arbitrary map $f$ from an abelian group $G$ to a set $X$, namely $K=\{g:\forall g':f(g'+g)=f(g')\}$. Thus $G/K_G$ is the largest quotient group of $G$ through which $f$ factors. This extends the usual kernel of quadratic maps on vector spaces. | |
| Feb 19, 2023 at 10:57 | history | edited | YCor | CC BY-SA 4.0 |
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| Feb 18, 2023 at 23:37 | history | edited | Bipolar Minds | CC BY-SA 4.0 |
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| Feb 18, 2023 at 23:35 | comment | added | Bipolar Minds | @LSpice Correct! | |
| Feb 18, 2023 at 23:32 | comment | added | LSpice | What property should the kernel $K$ have? The end of your first paragraph suggests you might want something like $K$ being minimal with respect to the requirement that there is a quadratic form $\bar q$ on $G/K$ that pulls back to $q$. Is that, together presumably with some sort of uniqueness, correct? (Also, a notational suggestion: old-fashioned texts often used $[\cdot]$ for the greatest-integer function, so $q(n) = n/2 + \mathbb Z$ might be clearer than $q(n) = [n/2]$.) | |
| Feb 18, 2023 at 21:33 | history | edited | Bipolar Minds | CC BY-SA 4.0 |
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| Feb 18, 2023 at 21:20 | history | asked | Bipolar Minds | CC BY-SA 4.0 |