# Regulator constants of integral representations of finite groups

## TL;DR

This paper studies regulator constants as invariants of integral group representations, establishing their properties and how they determine lattice isomorphism classes for certain finite groups, with implications for elliptic curves.

## Contribution

It introduces and analyzes the pairing between Brauer relations and the representation ring via regulator constants, proving non-degeneracy in specific cases and linking invariants to lattice classification.

## Key findings

- Pairing between Brauer relations and representation ring is never zero.
- Non-degeneracy of the pairing for groups with cyclic Sylow p-subgroups and permutation lattices.
- Lattice isomorphism classes are determined by regulator constants, scalar extension, and Yakovlev's cohomological invariant.

## Abstract

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_{p}[G]$-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser--Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p[G]$-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.