Chinner, Trinity2021-09-292021-09-292021-09-29http://hdl.handle.net/10393/42759http://dx.doi.org/10.20381/ruor-26976In this thesis, we classify up to conjugacy the maximal elliptic toral subgroups of all special orthogonal groups SO(V), where (q,V) is a 4-dimensional quadratic space over a non-archimedean local field of odd residual characteristic. Our parameterization blends the abstract theory of Morris with a generalization of the practical work performed by Kim and Yu for Sp(4). Moreover, we compute an explicit Witt basis for each such torus, thereby enabling its concrete realization as a set of matrices embedded into the group. This work can be used explicitly to construct supercuspidal representations of SO(V).enAttribution-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nd/4.0/Torusp-adicEllipticLocal fieldSpecial orthogonal groupNon-archimedeanThesis