By J. F. Davis
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Additional info for A Survey of Spherical Space Form Problem
Let E 2 (K) denote the squares in E(K). Let E+(K) denote the elements of E(K) which are positive at all real places. 11 Let K/Q be totally real and Galois of degree 2'. (a) E+(K) = E 2 (K) if and only if there is a u E E(K) such that, Nu = -1. ) (b) Let H be a subgroup of E (K) offinite index. If there is a u EH with Nu = -1, then IE(K)/ HI is odd. THEOREM Let An = 'n +,~I. 12 (~) = -1, If K is Kp' or K pq with the quadratic or the maximal 2 -extension of Q in Q[A 2 ", Apl with p ~ 1 (mod 8) then E 2 (K) = E+(K) and the cyclotomic units have odd index in U(K).
Let v E H with a> 0 minimal. If a were even, then v E H+ = H 2 and hence v 2 EH. This would contradict the minimality of a. Hence IE(K)/ HI is odd. 14  (a) Letp==3(4) then (T4(Q(8p,q,I))=Ojorqprime if and only if (i) q== 1(8) or (ii) q == 5(8) but p" == ± I (mod q) jor some odd v. (b) (T4(Q(8p, I, 1))=0 if p==q==l(mod4) but (~)=-1. In particular, (T4( Q(24, 5, I)) ;£. O. But (T4( Q(24, 13, I)) = O. When n > 3 there are several major differences. First, Q(2 n , p, I) is not a subgroup of the unit quaternions S3, while Q(8, p, I) = Q(8p) is.
29 Here Zp = lim Z/ pn is the p-adic completion of Z and Qp is its quotient field. For the arithmetic and geometric properties of these rings see , . 30 ;, U Kt(QpG) ~ Ko(ZG) pllGI ~ K o( zC ~I) G) Et) PrJ,1 Ko(ZpG) ~ U Ko(QpG). pllGI PERIODIC RESOLUTIONS 247 See Milnor  for the definition of the boundary map. 29. Suppose we are given an exact sequence over ZpG for all p dividing 1GI with the Cj free. Suppose for fixed j, the ranks over tpG are equal for all p. (Actually these sequences are very easy to construct for p-hyperelementary P-groups [31, 57]).
A Survey of Spherical Space Form Problem by J. F. Davis