Non-commutative Iwasawa theory for modular forms

TL;DR

This paper provides numerical evidence supporting the non-commutative Iwasawa main conjecture for motives of primitive modular forms over certain Galois extensions, confirming complex theoretical predictions.

Contribution

It offers the first extensive numerical verification of the non-commutative Iwasawa main conjecture for modular forms, including intricate cases with multiple critical points.

Findings

Numerical data aligns with the conjecture's predictions.

Supports Kato's congruence between different p-adic L-functions.

Confirms the conjecture in non-abelian Galois extension contexts.

Abstract

The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kato's mysterious congruence between the cyclotomic Manin p-adic L-function, and the cyclotomic p-adic L-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.