EXTENSIONS OF TORI IN SL ( 2 ) Jeffrey

Let ̃ SL(2, F) be the metaplectic two-fold cover of SL(2, F), the special linear group in two variables over a local field F of characteristic 0. The inverse image T̃ of a maximal torus T in ̃ SL(2, F) is an abelian extension of T by ±1. We consider the question of whether this extension is trivial. More generally we find the minimal subgroup A of the circle for which the extension is split when considered with coefficients in A. We see that |A| = 2, 4 or 8 in the p-adic case. We also find an explicit splitting function for the cocycle.