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.

David, C., Pappalardi, F. (2004). Average Frobenius distribution for inerts in \$Bbb Q(i)\$. JOURNAL OF THE RAMANUJAN MATHEMATICAL SOCIETY, 19, 181-201.

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

#### Abstract

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.
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2004
David, C., Pappalardi, F. (2004). Average Frobenius distribution for inerts in \$Bbb Q(i)\$. JOURNAL OF THE RAMANUJAN MATHEMATICAL SOCIETY, 19, 181-201.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11590/151040`
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