Let E be an elliptic curve over Q.Let a_(p) denote the trace of the Frobenius endomorphism at a rational prime p.For a fixed integer r,define the prime-counting function asπE,r(x):=Σp≤x,p∤ΔE,a_(p)=r^(1.)The Lang-T...Let E be an elliptic curve over Q.Let a_(p) denote the trace of the Frobenius endomorphism at a rational prime p.For a fixed integer r,define the prime-counting function asπE,r(x):=Σp≤x,p∤ΔE,a_(p)=r^(1.)The Lang-Trotter conjecture predicts thatπE,r(x)=C_(E,r)·√x/logx+o(√x/logx)as x→∞,where C_(E,r) is a specific non-negative constant.The Hardy-Littlewood conjecture gives a similar asymptotic formula as above for the number of primes of the form ax^(2)+bx+c.Assuming that the Hardy-Littlewood conjecture holds,we determine the constant CED,r for ED:y^(2)=x^(3)+Dx.As a consequence,we establish a relationship between the Hardy-Littlewood conjecture and the Lang-Trotter conjecture for the elliptic curve y^(2)=x^(3)+Dx.We show that the Hardy-Littlewood conjecture implies the Lang-Trotter conjecture for y^(2)=x^(3)+Dx.Conversely,if the Lang-Trotter conjecture holds for some D and 2r(for y^(2)=x^(3)+Dx,p∤D,a_(p) is always even)with the positive constant CED,r,then the polynomial x^(2)+r^(2) represents infinitely many primes.For a prime p,if a_(p)=2r,then p is necessarily of the form x^(2)+r^(2).Fixing r and D,and assuming that the Hardy-Littlewood conjecture holds,we obtain the density of the primes with a_(p)=2r inside the set of primes of the form x^(2)+r^(2).In some cases,the density is 1/4,which aligns with natural expectations,but this does not hold for all D.In particular,we give a full list of D and r when there is no prime p for a_(p)=2r.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.12231009 and 11971224)the Ministry of Science and Technology of China(Grant No.2020YFA0713800).
文摘Let E be an elliptic curve over Q.Let a_(p) denote the trace of the Frobenius endomorphism at a rational prime p.For a fixed integer r,define the prime-counting function asπE,r(x):=Σp≤x,p∤ΔE,a_(p)=r^(1.)The Lang-Trotter conjecture predicts thatπE,r(x)=C_(E,r)·√x/logx+o(√x/logx)as x→∞,where C_(E,r) is a specific non-negative constant.The Hardy-Littlewood conjecture gives a similar asymptotic formula as above for the number of primes of the form ax^(2)+bx+c.Assuming that the Hardy-Littlewood conjecture holds,we determine the constant CED,r for ED:y^(2)=x^(3)+Dx.As a consequence,we establish a relationship between the Hardy-Littlewood conjecture and the Lang-Trotter conjecture for the elliptic curve y^(2)=x^(3)+Dx.We show that the Hardy-Littlewood conjecture implies the Lang-Trotter conjecture for y^(2)=x^(3)+Dx.Conversely,if the Lang-Trotter conjecture holds for some D and 2r(for y^(2)=x^(3)+Dx,p∤D,a_(p) is always even)with the positive constant CED,r,then the polynomial x^(2)+r^(2) represents infinitely many primes.For a prime p,if a_(p)=2r,then p is necessarily of the form x^(2)+r^(2).Fixing r and D,and assuming that the Hardy-Littlewood conjecture holds,we obtain the density of the primes with a_(p)=2r inside the set of primes of the form x^(2)+r^(2).In some cases,the density is 1/4,which aligns with natural expectations,but this does not hold for all D.In particular,we give a full list of D and r when there is no prime p for a_(p)=2r.
