Kida’s Formula and Congruences

Let f be a modular eigenform of weight at least two and let F be a finite abelian extension of Q. Fix an odd prime p at which f is ordinary in the sense that the p Fourier coefficient of f is not divisible by p. In Iwasawa theory, one associates two objects to f over the cyclotomic Zp-extension F∞ of F : a Selmer group Sel(F∞, Af ) (whereAf denotes the divisible version of the two-dimensional Galois representation attached to f) and a p-adic L-function Lp(F∞, f). In this paper we prove a formula, generalizing work of Kida and Hachimori–Matsuno, relating the Iwasawa invariants of these objects over F with their Iwasawa invariants over p-extensions of F . For Selmer groups our results are significantly more general. Let T be a lattice in a nearly ordinary p-adic Galois representation V ; set A = V/T . When Sel(F∞, A) is a cotorsion Iwasawa module, its Iwasawa μ-invariant μ(F∞, A) is said to vanish if Sel(F∞, A) is cofinitely generated and its λ-invariant λ (F∞, A) is simply its p-adic corank. We prove the following result relating these invariants in a p-extension.