Transition formulae for ranks of abelian varieties

Abstract

Let A/k denote an abelian variety defined over a number field k with good ordinary reduction at all primes above p, and let K∞ =∪n≥1 Kn be a p-adic Lie extension of k containing the cyclotomic Zp-extension. We use K-theory to find recurrence relations for the λ-invariant at each σ-component of the Selmer group over K∞, where σ : Gk → GL(V ). This provides upper bounds on the Mordell-Weil rank for A(Kn) as n → ∞ whenever G∞ = Gal(K∞/k) has dimension at most 3.