[Paper Review] Asymptotics of conductors of elliptic curves over Q

Sean Howe, Kirti Joshi|arXiv (Cornell University)|Jan 22, 2012

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper investigates the asymptotic distribution of conductors of elliptic curves over ℚ, proving under the Hardy-Littlewood conjecture that there are infinitely many prime conductors, and unconditionally showing infinitely many almost prime and semistable conductors via Iwaniec's theorem. Assuming a strong Cohen-Lenstra heuristic, it bounds the upper density of prime conductors at approximately 0.686, suggesting most primes are not conductors.

ABSTRACT

In this note we study numbers which occur as conductors of elliptic curves over Q. We show, by constructing families of elliptic curves with quadratic discriminant and invoking a theorem of Iwaniec, that this set contains infinitely many almost primes. We show, assuming a strong version of the Cohen-Lenstra heuristics, that the set of prime conductors has an explicitly bounded density in the set of primes. Studying the Cremona and Stein-Watkins databases of elliptic curves we conjecture that the set of conductors should be of density zero in the set of natural numbers and that the set of prime conductors should be of density zero in the set of prime numbers.

Motivation & Objective

• To understand the asymptotic density and distribution of conductors of elliptic curves over ℚ.
• To determine whether the set of conductors has positive density in ℕ or in the set of primes.
• To investigate the infinitude and density of prime and almost prime conductors using analytic number theory and heuristic models.
• To conjecture that conductors have density zero in ℕ and in ℙ, based on data from Cremona and Stein-Watkins databases.
• To explore the role of 2-torsion and class group statistics in constraining conductor existence via Cohen-Lenstra-type heuristics.

Proposed method

• Constructs infinite families of elliptic curves with quadratic discriminants to generate conductors with few prime factors.
• Applies Iwaniec’s theorem on almost primes to unconditionally establish the infinitude of almost prime conductors.
• Uses Setzer’s classification of elliptic curves with rational 2-torsion and conductor p = u² + 64 to link conductor existence to primes of a specific form.
• Translates existence of curves with trivial 2-torsion into 3-divisibility conditions on class numbers of quadratic fields ℚ(√±p) and ℚ(√±2p).
• Applies a refined Cohen-Lenstra heuristic to quadratic fields with prime discriminants in specific congruence classes mod 8.
• Estimates upper density of prime conductors by combining class group statistics and independence assumptions in the heuristic model.

Experimental results

Research questions

• RQ1 Does the set of conductors of elliptic curves over ℚ have positive density in ℕ?
• RQ2 Are there infinitely many prime conductors for elliptic curves over ℚ?
• RQ3 What is the upper density of prime conductors within the set of all primes?
• RQ4 How do class group statistics of quadratic fields relate to the existence of elliptic curves with given conductors?
• RQ5 Can the distribution of conductors be modeled using refined Cohen-Lenstra heuristics for prime discriminants?

Key findings

• Assuming the Hardy-Littlewood conjecture, there are infinitely many prime conductors for elliptic curves over ℚ.
• Unconditionally, there are infinitely many almost prime conductors and infinitely many semistable conductors, via Iwaniec’s theorem on almost primes.
• Under a strong Cohen-Lenstra-type conjecture, the upper density of prime conductors in the set of all primes is at most approximately 0.686.
• The set of conductors has density zero in ℕ, based on numerical evidence from databases up to 10⁸ and heuristic reasoning.
• The set of prime conductors has density zero in the set of all primes, as supported by data and conditional on the Cohen-Lenstra heuristic.
• Consecutive non-conductor runs are unbounded, with data showing runs of length 45 and longer, suggesting the possibility of arbitrarily long gaps.