[Paper Review] Character sums for elliptic curve densities

Julio Brau|arXiv (Cornell University)|Mar 12, 2017

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper develops a character sum framework to compute entanglement correction factors for densities of primes $p$ for which the group of $ F_p$-rational points on an elliptic curve $E$ over $\mathbb{Q}$ is cyclic. By interpreting the correction factor as a character sum, the method enables explicit non-vanishing criteria and is applied to study densities under arithmetic progression conditions and in Koblitz's conjecture, yielding precise computational predictions for such densities.

ABSTRACT

If $E$ is an elliptic curve over $\mathbb{Q}$, then it follows from work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes $p$ such that the group of $\mathbb{F}_p$-rational points of the reduced curve $ ilde{E}(\mathbb{F}_p)$ is cyclic can be written as an infinite product $\prod δ_\ell$ of local factors $δ_\ell$ reflecting the degree of the $\ell$-torsion fields, multiplied by a factor that corrects for the entanglements between the various torsion fields. We show that this correction factor can be interpreted as a character sum, and the resulting description allows us to easily determine non-vanishing criteria for it. We apply this method in a variety of other settings. Among these, we consider the aforementioned problem with the additional condition that the primes $p$ lie in a given arithmetic progression. We also study the conjectural constants appearing in Koblitz's conjecture, a conjecture which relates to the density of primes $p$ for which the cardinality of the group of $\mathbb{F}_p$-points of $E$ is prime.

Motivation & Objective

• To develop a systematic method for computing entanglement correction factors in the density of primes $p$ for which $\tilde{E}(\mathbb{F}_p)$ is cyclic.
• To generalize Hooley's approach for Artin's primitive root conjecture to the setting of elliptic curve point group structures.
• To provide explicit non-vanishing criteria for the density constants in the cyclic group case, particularly under GRH.
• To extend the framework to study densities of primes in arithmetic progressions and to analyze Koblitz's conjecture on the primality of $\#\tilde{E}(\mathbb{F}_p)$.

Proposed method

• The entanglement correction factor $\mathfrak{C}_E$ is interpreted as a character sum, enabling explicit computation and non-vanishing analysis.
• The method uses Galois-theoretic decomposition of torsion fields $\mathbb{Q}(E[\ell])$ and their intersections via the Galois group $\mathrm{Gal}(\mathbb{Q}(E[m]) / \mathbb{Q})$.
• Local densities $\delta_\ell = 1 - \frac{1}{[\mathbb{Q}(E[\ell]):\mathbb{Q}]}$ are combined with the character sum correction to form the global density $C_E = \mathfrak{C}_E \prod_\ell \delta_\ell$.
• For curves with $\mathbb{Q}(E[2]) \subset \mathbb{Q}(E[3])$, the correction factor is computed via the genus-0 modular curve $X_H$, whose rational points parametrize such curves.
• The method leverages machine computation to verify the density prediction against empirical data for $p \leq 10^8$, achieving high numerical agreement.
• The framework is extended to study primes in arithmetic progressions and to analyze Koblitz's conjecture by computing the relevant correction factors via character sums.

Experimental results

Research questions

• RQ1 What is the precise form of the entanglement correction factor for the density of primes $p$ such that $\tilde{E}(\mathbb{F}_p)$ is cyclic, and when does it vanish?
• RQ2 How can the correction factor be expressed as a character sum to allow for effective non-vanishing criteria?
• RQ3 Can the method be adapted to compute densities of such primes when restricted to arithmetic progressions?
• RQ4 What is the structure of non-abelian intersections between torsion fields $\mathbb{Q}(E[m_1])$ and $\mathbb{Q}(E[m_2])$ for coprime $m_1, m_2$?
• RQ5 How does the method apply to Koblitz's conjecture on the density of primes for which $\#\tilde{E}(\mathbb{F}_p)$ is prime?

Key findings

• The entanglement correction factor $\mathfrak{C}_E$ is shown to be expressible as a character sum, enabling explicit computation and non-vanishing analysis.
• For the elliptic curve $Y^2 = X^3 - 63504X + 6223392$, the predicted density of primes with cyclic $\tilde{E}(\mathbb{F}_p)$ is $C_E \approx 0.831066$, matching the empirical density of $0.831069$ up to $p \leq 10^8$.
• The correction factor vanishes if and only if the character sum evaluates to zero, providing a clear criterion for when the density is zero.
• The method successfully extends to the case of primes in arithmetic progressions, generalizing the classical Artin-type density results.
• The only non-abelian entanglement between torsion fields occurs when $\mathbb{Q}(E[2]) \subset \mathbb{Q}(E[3])$, and such curves are parametrized by a genus-0 modular curve $X_H$ with explicit $j$-invariant $j = 2^{10}3^3 t^3(1 - 4t^3)$.
• The framework provides a systematic way to compute correction factors for Koblitz's conjecture and similar problems by reducing them to character sum evaluations.