Timeline for Characters of simply connected semsimple algebraic groups over local fields
Current License: CC BY-SA 4.0
16 events
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| Nov 18, 2018 at 14:31 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
Mistake corrected following comments of Arkandias.
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| Nov 16, 2018 at 20:00 | comment | added | Mikhail Borovoi | @Arkandias: The maps $G(k)\to\mathbb{Z}_2/2\mathbb{Z}_2$ that I constructed is not a homomorphism! I will edit my answer tomorrow. | |
| Nov 16, 2018 at 19:47 | comment | added | Arkandias | Finally, this subgroup has cardinal $(q^n-1)/(q-1)$ where $n^2$ is the dimension of $A$ over $k$. So in your case ($k=\mathbb{Q}_2$ and $A$ is the quaternion algebra), the order of $G_0/H_1$ should be $3$ according to this corollary, so in particular there should not be any morphism into $\mathbb{Z}/2\mathbb{Z}$. Yet, I do not see anything wrong with your construction... | |
| Nov 16, 2018 at 19:42 | comment | added | Arkandias | $k$ is a $p$-adic field with ring of integers $\mathfrak{o}$ and maximal ideal $\mathfrak{p}$, $A$ is a central division algebra over $k$ with ring of integers $\mathfrak{D}$ and maximal ideal $\mathfrak{P}$. The corollary claims that the commutator of $G_0 := \mathrm{SL}(1,A)$ is the congruence subgroup $H_1$ consisting of those elements $g \in G_0$ which are $\equiv 1 \bmod \mathfrak{P}$. So the abelianization is isomorphic to $G_0/H_1$, which in turn is isomorphic to the norm $1$ subgroup of the field extension $\mathfrak{D}/\mathfrak{P}$ of $\mathfrak{o}/\mathfrak{p}$. | |
| Nov 16, 2018 at 13:45 | comment | added | Mikhail Borovoi | @Arkandias: Do I understand correctly that in our case $q=2$, so $q+1=3$ ? | |
| Nov 16, 2018 at 13:44 | comment | added | Mikhail Borovoi | @Arkandias: Could you please explain how the corollary to Theorem 21 implies that the abelianization of ${\rm SL}(1,D)$ has order $q+1$? | |
| Nov 15, 2018 at 21:36 | comment | added | Arkandias | Yes, but the Corollary includes that case (see just above the statement, and most of the proof is about the dyadic quaternion algebra case). | |
| Nov 15, 2018 at 18:58 | comment | added | Mikhail Borovoi | @Arkandias: In Theorem 21 Riehm assumes that $D$ is not a dyadic quaternion algebra. | |
| Nov 15, 2018 at 16:02 | comment | added | Arkandias | This example seems to contradict the Corollary to Theorem 21 in Riehm "The norm 1 group of a $p$-adic division algebra", which implies that the abelianization of $\mathrm{SL}(1,D)$ has order $q+1$. Am I missing something? | |
| May 20, 2016 at 9:29 | vote | accept | Daniel Loughran | ||
| May 20, 2016 at 9:27 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| May 20, 2016 at 5:19 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| May 19, 2016 at 20:19 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| May 19, 2016 at 20:06 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| May 19, 2016 at 19:50 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| May 19, 2016 at 19:44 | history | answered | Mikhail Borovoi | CC BY-SA 3.0 |