Almost prime values of the order of elliptic curves over finite fields
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Abstract
Abstract. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> </m:math> $E$ be an elliptic curve over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi/> </m:math> ${\mathbb {Q}}$ without complex multiplication. For each prime <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> $p$ of good reduction, let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>E</m:mi> <m:mo>(</m:mo> <m:msub> <m:mi/> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> <m:mo>|</m:mo> </m:mrow> </m:math> $|E({\mathbb {F}}_p)|$ be the order of the group of points of the reduced curve over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi/> <m:mi>p</m:mi> </m:msub> </m:math> ${\mathbb {F}}_p$ . According to a conjecture of Koblitz, there should be infinitely many such primes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> $p$ such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>E</m:mi> <m:mo>(</m:mo> <m:msub> <m:mi/> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> <m:mo>|</m:mo> </m:mrow> </m:math> $|E({\mathbb {F}}_p)|$ is prime, unless there are some local obstructions predicted by the conjecture. Suppose that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> </m:math> $E$ is a curve without local obstructions (which is the case for most elliptic curves over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi/> </m:math> ${\mathbb {Q}}$ ). We prove in this paper that, under the GRH, there are at least <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>.</m:mo> <m:mn>778</m:mn> <m:msubsup> <m:mi>C</m:mi> <m:mi>E</m:mi> <m:mi>twin</m:mi> </m:msubsup> <m:mi>x</m:mi> <m:mo>/</m:mo> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mo form="prefix">log</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> $2.778 C_E^{\rm twin} x / (\log x)^2$ primes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> $p$ such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>E</m:mi> <m:mo>(</m:mo> <m:msub> <m:mi/> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> <m:mo>|</m:mo> </m:mrow> </m:math> $|E({\mathbb {F}}_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>C</m:mi> <m:mi>E</m:mi> <m:mi>twin</m:mi> </m:msubsup> </m:math> $C_E^{\rm twin}$ appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
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| Bibliometrics | 0.000 | 0.000 |
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| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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