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PreprintCC BY 4.0arXiv

Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve

  • SeungJu Lee
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Abstract

Let E:y2=x3−x. We prove that #E(Fp)/8 is squarefree and has at most two prime factors for ≫ X/(log X)2 primes p≤ X. The proof follows the Gaussian CM sieve framework of Iwaniec and Jiménez Urroz and incorporates the intersection terms identified by Xie. For a squarefree Gaussian ideal a, we separate its largest rational ideal factor and use the resulting divisibility of both integral coordinates to control the large-factor tail. This transfers an ideal Bombieri--Vinogradov estimate to rational norm divisibility at level 1/2, with any fixed power of the divisor function as a weight. We prove the corresponding distribution estimate for the switched product sequence and apply the transfer to both sequences in Chen's weighted sieve. The primes counted may be restricted to p≡5pmod8; their normalized orders are odd, squarefree, and supported on primes congruent to 1pmod4.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026divisibility,
  title = {{Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve}},
  author = {SeungJu Lee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20010},
  eprint = {2610.20010},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20010},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve
AU  - SeungJu Lee
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20010
UR  - https://arxiv.org/abs/2610.20010
AB  - Let E:y^2=x^3−x. We prove that #E(F_p)/8 is squarefree and has at most two prime factors for ≫ X/(log X)^2 primes p≤ X. The proof follows the Gaussian CM sieve framework of Iwaniec and Jiménez Urroz and incorporates the intersection terms identified by Xie. For a squarefree Gaussian ideal a, we separate its largest rational ideal factor and use the resulting divisibility of both integral coordinates to control the large-factor tail. This transfers an ideal Bombieri--Vinogradov estimate to rational norm divisibility at level 1/2, with any fixed power of the divisor function as a weight. We prove the corresponding distribution estimate for the switched product sequence and apply the transfer to both sequences in Chen's weighted sieve. The primes counted may be restricted to p≡5pmod8; their normalized orders are odd, squarefree, and supported on primes congruent to 1pmod4.
ER  -