Numerical evidence toward a 2-adic equivariant "main conjecture"

TL;DR

This paper provides numerical evidence supporting a 2-adic equivariant main conjecture in Iwasawa theory, linking it to specific congruence conditions on power series coefficients for dihedral extensions.

Contribution

It establishes an equivalence between the conjecture and a congruence condition, and verifies this condition in numerous cases for dihedral extensions of order 8.

Findings

The conjecture is equivalent to a specific 2-adic congruence condition.

Numerical verification confirms the congruence holds in many examples.

Supports the conjecture's validity in the tested cases.

Abstract

Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that, for the prime 2 and a dihedral extension of order 8 over Q, this conjecture is equivalent to a congruence condition on the coefficients of a power series with 2-adic integral coefficients constructed using the 2-adic L-series associated to the extension. We then verify that this congruence condition holds for the first coefficients in a large number of examples.