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Kolyvagin's conjecture at non-ordinary primes

  • Antonio Lei
  • Luochen Zhao
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Abstract

Let K be an imaginary quadratic field and let p β‰₯ 5 be a prime that is unramified in K. Let π’œf/β„š be an abelian variety of GL2-type associated with a weight-two modular form f, with good non-ordinary reduction at p, and suppose that (f,K) satisfies the generalized Heegner hypothesis. In the case where p is inert in K, we further assume that π’œf is an elliptic curve. We develop an Euler-characteristic formula for signed Selmer groups over anticyclotomic β„€p-extensions that applies when the corresponding Selmer modules have arbitrary Ξ›-rank. Assuming one inclusion in the signed Iwasawa main conjecture, we apply this formula to prove Kolyvagin's conjecture on the non-vanishing of the Kolyvagin system attached to Heegner points. Our results extend to the non-ordinary setting the results of Wei Zhang, Burungale--Castella--Grossi--Skinner, Castella--Sano and Kim in the ordinary case, and complement the works of Sweeting and Kim in the non-ordinary case under different hypotheses. We also study the effect of the exceptional zero phenomenon on the Iwasawa main conjecture in the inert case.

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Cite

BibTeXRIS
BibTeX
@misc{lei2026kolyvagins,
  title = {{Kolyvagin's conjecture at non-ordinary primes}},
  author = {Antonio Lei and Luochen Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13088},
  eprint = {2609.13088},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13088},
}
RIS
TY  - UNPB
TI  - Kolyvagin's conjecture at non-ordinary primes
AU  - Antonio Lei
AU  - Luochen Zhao
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.13088
UR  - https://arxiv.org/abs/2609.13088
AB  - Let K be an imaginary quadratic field and let p β‰₯ 5 be a prime that is unramified in K. Let π’œ_f/β„š be an abelian variety of GL_2-type associated with a weight-two modular form f, with good non-ordinary reduction at p, and suppose that (f,K) satisfies the generalized Heegner hypothesis. In the case where p is inert in K, we further assume that π’œ_f is an elliptic curve. We develop an Euler-characteristic formula for signed Selmer groups over anticyclotomic β„€_p-extensions that applies when the corresponding Selmer modules have arbitrary Ξ›-rank. Assuming one inclusion in the signed Iwasawa main conjecture, we apply this formula to prove Kolyvagin's conjecture on the non-vanishing of the Kolyvagin system attached to Heegner points. Our results extend to the non-ordinary setting the results of Wei Zhang, Burungale--Castella--Grossi--Skinner, Castella--Sano and Kim in the ordinary case, and complement the works of Sweeting and Kim in the non-ordinary case under different hypotheses. We also study the effect of the exceptional zero phenomenon on the Iwasawa main conjecture in the inert case.
ER  -