Comparison of different definitions of pseudocharacters

TL;DR

This paper proves the equivalence of different definitions of d-dimensional pseudocharacters over any ring and compares the scheme of Lafforgue's G-pseudocharacters with the G-character variety.

Contribution

It establishes the equivalence of Chenevier's and Lafforgue's definitions of pseudocharacters and analyzes the relationship between Lafforgue's G-pseudocharacters and the G-character variety.

Findings

Definitions of pseudocharacters by Chenevier and Lafforgue agree over any ring.

The scheme of Lafforgue's G-pseudocharacters is compared with the G-character variety.

The paper clarifies the structure and relationships of pseudocharacters and character varieties.

Abstract

We prove that the definitions of a $d$-dimensional pseudocharacter (or pseudorepresentation) given by Chenevier and V. Lafforgue agree over any ring. We also compare the scheme of Lafforgue's $G$-pseudocharacters of a group with its $G$-character variety.