The Lang-Trotter Conjecture for the elliptic curve $y^2=x^3+Dx$

TL;DR

This paper explores the relationship between the Hardy-Littlewood and Lang-Trotter conjectures for elliptic curves, establishing implications and analyzing prime distributions related to specific trace values.

Contribution

It demonstrates that the Hardy-Littlewood Conjecture implies the Lang-Trotter Conjecture for certain elliptic curves and vice versa, linking prime representations to elliptic curve trace distributions.

Findings

Hardy-Littlewood Conjecture implies Lang-Trotter Conjecture for $y^2=x^3+Dx$

If Lang-Trotter holds for some $D$ and $2r$, then $x^2+r^2$ represents infinitely many primes

Density of primes with $a_p=2r$ can be $1/4$ in some cases

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}.$ 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 as $\pi_{E,r}(x):=\sum_{p\leq x,p\nmid \Delta_E,a_p=r}1$. The Lang-Trotter Conjecture predicts that $$\pi_{E,r}(x)=C_{E,r}\cdot \frac{\sqrt{x}}{{\rm log}x}+o(\frac{\sqrt{x}}{{\rm log}x})$$ as $x\longrightarrow \infty,$ 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 $ax^2+bx+c$. 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 $y^2=x^3+Dx, p\nmid D, a_p$ is always even) with positive constant $C_{E,2r},$ 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 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 $a_p=2r$.