Power Residue Symbols and the Central Sections of SL(2, A)

In TheE(2,A) sections of SL(2,A) (Ann. of Math. 134 (1991), 159–188), we locate the E(A) normalized subgroups of SL(2,A) in central sections of SL(2,A) for all subrings of Q and all commutative rings satisfying SR ₂ In solving this problem we introduced the notion of radix (see (1.1)) and the group C(P_x) = E(2,A),E(2,A;P_x)] = SL(2,A), SL(2,A;P_x)] for the rings considered here.The purpose of this paper is to determine SL(2,A;P_xC(P_x) for SR ₂ rings and number rings with infinitely many units.In Section 2, Mennicke symbols for Jordan ideals are defined. They are determined for number rings and shown to be connected to power residue symbols in a delicate way. This extends the work of Bass, Milnor and Serre.In Section 3, an explicit homomorphism from E(2,Al;P_x) into an additive section of A is given for all commutative rings A. If A satisfiesSR ₂ the kernel of this map is C(P_x.The main problem for number rings is solved by giving an explicit homomorphism on SL(2,A;P_x) whose kernel is C(P_x).