Nonvanishing of 2-primary Kolyvagin classes via simultaneous quadratic twists

Abstract

Let $E/\mathbf{Q}$ be an elliptic curve, let $K=\mathbf{Q}(\sqrt{D})$ satisfy the Heegner hypothesis with $D<-4$, and fix a modular parametrization $φ$ with $φ(\infty)=O$. Under surjectivity modulo $2^L$, with $L\ge 2$, and a cohomological exclusion condition at level $4$, we construct a finite set of primes $T$ such that, writing $d_T=\prod_{q\in T}(-q)$ and $n_T=\prod_{q\in T}q$, 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 $v_2(m_T)=v_2(c_φ)+\mathfrak{i}(E,K)$. The optimal Manin constant theorem supplies a $2$-indivisible parametrization $φ_0$, for which $κ^{\mathrm{std}}_{n_T,1}(φ_0)\ne 0$ if and only if $\mathfrak{i}(E,K)=0$. 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 $1\le M\le L$. Full $2$-adic Galois image then gives strong nonvanishing.