Greenberg’s minimality conjecture for analytic ranks in Hida families

Establish Greenberg’s minimality conjecture by proving that, for a non-CM newform of even weight and trivial Nebentypus with analytic rank one, all but finitely many even-weight, trivial-Nebentypus specializations in the associated ordinary p-adic Hida family have analytic rank one.

Background

The paper studies whether analytic rank one can propagate from a higher-weight classical modular form to almost all compatible specializations in an ordinary p-adic Hida family. Greenberg’s minimality conjecture predicts that the analytic ranks of even-weight, trivial-Nebentypus specializations should be as small as permitted by their functional equations, with only finitely many exceptions.

The main theorem proves this prediction conditionally on injectivity of a particular p-adic Abel–Jacobi map and positive definiteness of certain archimedean height pairings. Thus, the general conjecture remains unresolved, while the paper establishes a conditional result in the rank-one setting.