Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols

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