Let E be an elliptic curve over Q. The modular symbols attached to E are key to the arithmetic behavior of E over abelian extensions of Q. This lecture will report on the conjectured statistical structure—and specifically on what we call the variance slopes—of the distributions defined by the values of modular symbols. We formulate a heuristic that suggests finiteness of Mordell-Weil rank of E over certain large fields.

This is joint work with Karl Rubin. This work was supported by NSF grant DMS-0968831.