Let $p>2$ be prime. By the classification of quadratic forms, there are $8$ pairwise non-equivalent isotropic orthogonal groups in $4$ variables. Is there a concrete classification of orthogonal groups of these groups? Could they be isomorphism? The analogous question for $3$ variables has been asked and answered in a previous question of mine ( $p$-adic analogues of $SO(3)$ ) but the method used there does not apply to forms with $4$ variables.
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asked Dec 16, 2015 at 0:34
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$\begingroup$ For non-degenerate quadratic spaces $(V, q)$ and $(V', q')$ of dimension $n \ge 3$ over a field $k$, any $k$-group isomorphism ${\rm{SO}}(q) \simeq {\rm{SO}}(q')$ arises from a conformal isometry $q \simeq q'$ that is unique up to $k^{\times}$-scaling on $V$. (Indeed, Hilbert 90 reduces this to the case $k=k_s$, so then we may assume $q' = q$ and use Dieudonne's theorem (or better: the Isomorphism Theorem for split reductive groups) that the natural map ${\rm{PGO}}(q) \rightarrow {\rm{Aut}}_{{\rm{SO}}(q)/k}$ is an isomorphism.) Thus, ${\rm{SO}}(q)$ "remembers" $q$ up to conformal isometry. $\endgroup$– nfdc23Commented Dec 16, 2015 at 5:26
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There is a concrete description of the various forms in a paper by Harris, Soudry, and Taylor, "l-adic representations associated to modular forms over imaginary quadratic fields," Inventiones, 1993. The paper is available here: https://eudml.org/doc/144109
