Kida's formula and congruences

TL;DR

This paper establishes a Kida-type formula relating p-adic Iwasawa invariants of modular eigenforms and elliptic curves over abelian p-extensions, extending classical results to broader contexts using congruences between modular forms.

Contribution

It generalizes Kida's formula to modular eigenforms and elliptic curves at supersingular primes, employing congruences and Galois representations.

Findings

Derived a formula linking Iwasawa invariants over different fields.

Extended Kida's formula to supersingular primes.

Provided methods applicable to a broad class of Galois representations.

Abstract

We prove a formula (analogous to that of Kida in classical Iwasawa theory and generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side our methods, which make use of congruences between modular forms, yield a Kida-type formula for a very general class of ordinary Galois representations. We are further able to deduce a Kida-type formula for elliptic curves at supersingular primes.