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Serre's cyclicity conjecture for elliptic curves

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Let E/QE/{\mathbb Q} be an elliptic curve, and for each prime pp of good reduction let E~(Fp)\widetilde{E}({\mathbb F}_p) denote the group of rational points of its reduction modulo pp. Serre's cyclicity conjecture. The group E~(Fp)\widetilde{E}({\mathbb F}_p) is cyclic for infinitely many primes pp if and only if EE contains a non-rational point of order 22. This prediction is part of the study of cyclic reductions of elliptic curves. The surrounding text records conditional density results and positivity results, but gives no resolution of this claimed equivalence.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Serre's cyclicity conjecture for elliptic curves

    Let E/QE/\mathbb{Q} be an elliptic curve, and let πEcyc⁡(x)\pi_E^{\operatorname{cyc}}(x) count primes of cyclic reduction for EE up to xx. Let CEcyc⁡≥0C_E^{\operatorname{cyc}}\geq 0 be the explicit constant defined by the source.

    Serre's cyclicity conjecture.

    πEcyc⁡(x)∼CEcyc⁡⋅xlog⁡x\pi_E^{\operatorname{cyc}}(x)\sim C_E^{\operatorname{cyc}}\cdot\frac{x}{\log x}

    as x→∞x\to\infty.

    Serre proposed this analogue of Artin's primitive root conjecture and proved it under the Generalized Riemann Hypothesis.

    source: Sung Min Lee, Jacob Mayle and Tian Wang, “Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves”, arXiv:2408.16641 (2024).

Sources & referencesView supporting material

Primary source

Yildirim Akbal and Ahmet Muhtar Guloglu, “Cyclicity of Elliptic Curves Modulo Primes in arithmetic progressions”, arXiv:1904.08296 (2020).

analytic-number-theorycyclic-groupscyclicityelliptic-curvesprimes

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