Nonvanishing of 2-primary Kolyvagin classes via simultaneous quadratic twists

Abstract

Let E/Q be an elliptic curve, let K=Q(√D) satisfy the Heegner hypothesis with D<−4, and fix a modular parametrization φ with φ(∞)=O. Under surjectivity modulo 2^L, with L≥ 2, and a cohomological exclusion condition at level $4$, we construct a finite set of primes T such that, writing d_T=∏_{q∈ T}(−q) and n_T=∏_{q∈ T}q, the $2$-Selmer dimensions of E^d_T and E^d_TD are $0$ and $1$. The support size is bounded in terms of the initial Selmer dimensions, independently of L. Inspired by the work of Tian and Tian--Yuan--Zhang on congruent numbers, we construct a point Q_T on the terminal twist and identify its Kummer class with the standard mod-$2$ Kolyvagin class. Using the low-corank $2$-converse and the $2$-primary Birch--Swinnerton-Dyer formula, we prove that its index m_T in the full Mordell--Weil lattice satisfies v₂(m_T)=v₂(c_φ)+i(E,K). The optimal Manin constant theorem supplies a $2$-indivisible parametrization φ₀, for which κ^std_{n_T,1}(φ₀)≠ 0 if and only if i(E,K)=0. When the intrinsic index vanishes, the minimum terminal support size is R=min{d₂(E),d₂(E^D)}, and the normalized standard classes have order 2^M for 1≤ M≤ L. Full $2$-adic Galois image then gives strong nonvanishing.

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Cite

BibTeX RIS

BibTeX

@misc{sun2026nonvanishing,
  title = {{Nonvanishing of 2-primary Kolyvagin classes via simultaneous quadratic twists}},
  author = {Zhiqi Sun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31631},
  eprint = {2610.31631},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31631},
}

RIS

TY  - UNPB
TI  - Nonvanishing of 2-primary Kolyvagin classes via simultaneous quadratic twists
AU  - Zhiqi Sun
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31631
UR  - https://arxiv.org/abs/2610.31631
AB  - Let E/Q be an elliptic curve, let K=Q(√D) satisfy the Heegner hypothesis with D<−4, and fix a modular parametrization φ with φ(∞)=O. Under surjectivity modulo 2^L, with L≥ 2, and a cohomological exclusion condition at level $4$, we construct a finite set of primes T such that, writing d_T=∏_q∈ T(−q) and n_T=∏_q∈ Tq, the $2$-Selmer dimensions of E^d_T and E^d_TD are $0$ and $1$. The support size is bounded in terms of the initial Selmer dimensions, independently of L. Inspired by the work of Tian and Tian--Yuan--Zhang on congruent numbers, we construct a point Q_T on the terminal twist and identify its Kummer class with the standard mod-$2$ Kolyvagin class. Using the low-corank $2$-converse and the $2$-primary Birch--Swinnerton-Dyer formula, we prove that its index m_T in the full Mordell--Weil lattice satisfies v_2(m_T)=v_2(c_φ)+i(E,K). The optimal Manin constant theorem supplies a $2$-indivisible parametrization φ_0, for which κ^std_n_T,1(φ_0)≠ 0 if and only if i(E,K)=0. When the intrinsic index vanishes, the minimum terminal support size is R=min{d_2(E),d_2(E^D)}, and the normalized standard classes have order 2^M for 1≤ M≤ L. Full $2$-adic Galois image then gives strong nonvanishing.
ER  -

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