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YCor
YCor
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$p$-adic analogues of $SO$\mathrm{SO}(3)$

I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (non-degenerate) quadratic forms on the $3$-dimensional $p$-adic vector space ${\mathbb Q}_p^3$, one compact and one non-compact. The reference provided is Serrs'sSerre's "a course in arithmetic", Chapter IV, in which I was not able to find a proof. As far as I know, up to equivalence and scaling, there are three anisotropic and three isotropic such quadratic forms. Is it obvious that the corresponding orthogonal groups in each class are isomorphic?

$p$-adic analogues of $SO(3)$

I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (non-degenerate) quadratic forms on the $3$-dimensional $p$-adic vector space ${\mathbb Q}_p^3$, one compact and one non-compact. The reference provided is Serrs's "a course in arithmetic", Chapter IV, in which I was not able to find a proof. As far as I know, up to equivalence and scaling, there are three anisotropic and three isotropic such quadratic forms. Is it obvious that the corresponding orthogonal groups in each class are isomorphic?

$p$-adic analogues of $\mathrm{SO}(3)$

I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (non-degenerate) quadratic forms on the $3$-dimensional $p$-adic vector space ${\mathbb Q}_p^3$, one compact and one non-compact. The reference provided is Serre's "a course in arithmetic", Chapter IV, in which I was not able to find a proof. As far as I know, up to equivalence and scaling, there are three anisotropic and three isotropic such quadratic forms. Is it obvious that the corresponding orthogonal groups in each class are isomorphic?

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Keivan Karai
Keivan Karai
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$p$-adic analogues of $SO(3)$

I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (non-degenerate) quadratic forms on the $3$-dimensional $p$-adic vector space ${\mathbb Q}_p^3$, one compact and one non-compact. The reference provided is Serrs's "a course in arithmetic", Chapter IV, in which I was not able to find a proof. As far as I know, up to equivalence and scaling, there are three anisotropic and three isotropic such quadratic forms. Is it obvious that the corresponding orthogonal groups in each class are isomorphic?