观点

Theorem 4.1 ([The fixed-\(K\) missing-vertex implication]). Fix the ring data E,K , with corank_ ℤ_2 Sel_ 2^∞ (E/K)=1. Suppose binary families exist for arbitrarily large b , and for every nonzero x the product L(E^ (h_x) ,s)L(E^ (h_xk) …

Theorem 4.1 ([The fixed-K missing-vertex implication]). Fix the ring data E,K , with corank_ℤ2 Sel₂^∞ (E/K)=1. Suppose binary families exist for arbitrarily large b , and for every nonzero x the product L(E^ (h_x) ,s)L(E^ (h_xk) ,s) has a simple zero and 2j_hx -4w(h_x)-s(h_x)-s(h_xk)≤B with B independent of the family. Then ord_s=1 (L(E,s)L(E^ (k) ,s))=1. This implication does not assume Mordell--Weil rank one or finite Shafarevich--Tate group at the base.