Toward an explicit eigencurve for GL(3)

Abstract

Starting with a numerically noncritical (at [Formula: see text]) Hecke eigenclass [Formula: see text] in the homology of a congruence subgroup [Formula: see text] of [Formula: see text] (where [Formula: see text] divides the level of [Formula: see text]) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of [Formula: see text] to a Hecke eigenclass [Formula: see text] with coefficients in a module of [Formula: see text]-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection [Formula: see text] to weight space of the eigencurve around the point [Formula: see text] corresponding to the system of Hecke eigenvalues of [Formula: see text]. We do this under the conjecturally mild hypothesis that [Formula: see text] is smooth at [Formula: see text].