Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols

Suppose E is an elliptic curve over Q and χ a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1)=0. Via Birch Swinnerton-Dyer conjecture, this gives heuristic Mordell–Weil rank grows in abelian extensions Q. Using heuristic, we find large class infinite F where expect E(F) be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due David, Fearnley, Kisilevsky. Where our predictions theirs overlap, are consistent.