2021/08/13 by Hourong Qin, Qin, Hourong
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2108.06292
openalex publication_date 2021/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E be an elliptic curve over ℚ. Let ap denote the trace of the Frobenius endomorphism at a rational prime p. For a fixed integer r, define the prime-counting function as πE,r(x):=∑p≤ x,p\nmid ΔE,ap=r1. The Lang-Trotter Conjecture predicts that πE,r(x)=CE,r⋅ \frac√(x)\rm logx+o(\frac√(x)\rm logx) as x\longrightarrow ∞, where CE,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 ax2+bx+c. We establish a relationship between the Hardy-Littlewood Conjecture and the Lang-Trotter Conjecture for the elliptic curve y2=x3+Dx. We show that the Hardy-Littlewood Conjecture implies the Lang-Trotter Conjecture for y2=x3+Dx. Conversely, if the Lang-Trotter Conjecture holds for some D and 2r (for y2=x3+Dx, p\nmid D, ap is always even) with positive constant CE,2r, then the polynomial x2+r2 represents infinitely many primes. For a prime p, if ap=2r, then p is necessarily of the form x2+r2. Fixing r and D, and assuming that the Hardy-Littlewood Conjecture holds, we obtain the density of the primes with ap=2r inside the set of primes of the form x2+r2. In some cases, the density is 1/4, which is a natural expectation, but it fails to be true for all D. In particular, we give a full list of D and r when there is no prime p for ap=2r.