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PreprintCC BY 4.0arXiv

On Greenberg's conjecture for rational elliptic curves at Eisenstein primes

  • Zichao Lin
  • Mulun Yin
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Abstract

Let E/ℚ be an elliptic curve, and let p be an odd prime of ordinary reduction for E, and assume that E admits a rational p-isogeny. In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa μ-invariants of the Selmer groups attached to E over the cyclotomic ℤp-extension of the rational numbers. Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens's conjecture on the behavior of analytic μ-invariants.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026greenbergs,
  title = {{On Greenberg's conjecture for rational elliptic curves at Eisenstein primes}},
  author = {Zichao Lin and Mulun Yin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40282},
  eprint = {2609.40282},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40282},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Greenberg's conjecture for rational elliptic curves at Eisenstein primes
AU  - Zichao Lin
AU  - Mulun Yin
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40282
UR  - https://arxiv.org/abs/2609.40282
AB  - Let E/ℚ be an elliptic curve, and let p be an odd prime of ordinary reduction for E, and assume that E admits a rational p-isogeny. In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa μ-invariants of the Selmer groups attached to E over the cyclotomic ℤ_p-extension of the rational numbers. Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens's conjecture on the behavior of analytic μ-invariants.
ER  -