SL2 — The negative solutions

In the previous chapter, we dealt with solvable groups and the methods came from abelian groups. In this chapter, we deal with <math display='block'> <mrow> <mi>S</mi><msub> <mi>L</mi> <mn>2</mn> </msub> <mo stretchy='false'>(</mo><msub> <mi mathvariant='script'>O</mi> <mi>S</mi> </msub> <mo stretchy='false'>)</mo> </mrow> </math>$$S{L_2}({\mathcal{O}_S})$$ where <math display='block'> <mi mathvariant='script'>O</mi> </math>$$\mathcal{O}$$ is the ring of integers in an algebraic number field and S is any set of places containing all the archimedean ones. The methods here will be very different from those of the previous chapter.