Average Frobenius distribution for inerts in $Bbb Q(i)$

The authors of the paper under review extend their study of the Lang-Trotter conjecture for elliptic curves [Internat. Math. Res. Notices 1999, no. 4, 165--183; MR1677267 (2000g:11045)] to the generalised Lang-Trotter conjecture for elliptic curves, as formulated on page 182 of the paper under review. More precisely, they study the function 1#C∑E∈Cπr,fE(x), where C is a suitable set of elliptic curves E which may depend on x, #C is the cardinality of the set C, and πr,fE(x)=#{p:N(p)≤x,degK(p)=f, and ap(E)=r}, where E is an elliptic curve over a number field K/Q with a minimal model over the ring of integers OK, r is any fixed integer, f is a divisor of the degree [K:Q] of K over Q, p is a prime ideal in the ring of integers OK, N(p)=pf is the number of elements of the finite field OK/p, the degree of p over K is degK(p)=f, and the trace of Frobenius ap(E)=#Ep(OK/p)−N(p)−1 satisfies the Hasse bound |ap(E)|≤2N(p)−−−−√=2pf/2. The main result of the paper under review is that for any given nonzero integer r, 1#Cx∑E∈Cxπr,2E(x)∼crloglogx, where K=Q(i), Cx is the set of elliptic curves E:Y2=X3+αX+β, with α=a1+a2i∈Z[i], β=b1+b2i∈Z[i], and max{|a1|,|a2|,|b1|,|b2|}≤xlogx, and cr=13π∏l>2l(l−1−(−r2l))(l−1)(l−(−1l))=1.07820…3π. Furthermore, if r=0, then 1#Cx∑E∈Cxπ0,2E(x)<∞. This result is in agreement with the generalised Lang-Trotter conjecture in the case f=2, which asserts that there exists a positive constant CE,r,f such that πr,2E(x)∼CE,r,2loglogx. The main ingredients in the proof of the main result are rapidly decreasing functions and contour integration as found in the work by A. Akbary, David and the reviewer [Acta Arith. 111 (2004), no. 3, 239--268; MR2039225 (2004k:11086)], coupled with the framework outlined by the authors of the paper under review in [op. cit.]. The authors remark that a similar result should hold for inerts in a general imaginary quadratic field, but that it is not clear how to generalise the technical details of their proof to this general case.