Comparison of Different Definitions of Pseudocharacter

Abstract

Pseudocharacters

were first introduced by Taylor

and Wiles

to study congruences of representations.

When $A$ is an algebraically closed field with

$d! \in A^\times$,

the trace gives a bijection between

the set of congruence classes of semisimple $A$-valued representations and

the set of $A$-valued $d$-dimensional pseudocharacters.

However, in small characteristic this definition of pseudocharacter works less well.

Chenevier later defined determinants,

which are equivalent to the original defintion of pseudocharacter in characteristic zero,

but moreover, $A$-valued determinants correspond uniquely to congruence classes of $A$-valued semisimple representations for any algebraically closed field $A$.

Lafforgue gives a more general defintion of pseudocharacter that makes sense

for representations with values in any reductive algebraic group

(and in any characteristic).

This work shows that Chenevier's defintion of determinants is equivalent to Lafforgue's defintion (for $GL_d$) over any ring.

Another traditional approach to studying congruence classes of semisimple representations is to study the character variety. This work compares the space of Lafforgue pseudorepresentations to the character variety, proving that they are almost isomorphic (in a precise sense).

In particular, it shows that they have the same points with values in any perfect field.