Stabilizers of simple paths in the Bruhat-Tits tree of SL(2) over finite   extensions of Q2
  For F an algebraic extension of Q2, the conjugacy classes of invertible, 2-by-2, trace-zero matrices under the action of G := SL2(F) are analyzed relative to the quadratic extension that splits the respective characteristic polynomial. The stabilizer in G of each such matrix is computed as a stabilizer of a simple, Galois invariant path in the Bruhat-Tits Tree of G. 
Terence Joseph Kivran-Swaine
math.NT math.RT