# A Note on the Main Conjecture over Q

## TL;DR

This paper explores the connection between the main conjecture of Iwasawa theory over Q and Galois representations on the etale pro-p fundamental group, highlighting the role of Vandiver's conjecture.

## Contribution

It situates the main conjecture within the framework of Galois representations on the etale pro-p fundamental group, offering a new perspective.

## Key findings

- Shows the main conjecture's natural placement in Galois representation context
- Identifies the necessity of Vandiver's conjecture for proof
- Provides a conceptual link between Iwasawa theory and fundamental groups

## Abstract

In this note we show how the main conjecture of the Iwasawa theory over Q has a natural place in the context of the Galois representation of the Galois group $Gal(\bar Q/Q)$ on the etale pro-p fundamental group of the projective line minus three points. However we still need to assume the Vandiver conjecture to get a proof of the main conjecture in this context.