Computing the homology of SL3(Z) with infinite-dimensional coefficient modules

A. Ash, D. Pollack

Let p be a prime and Γ a congruence subgroup of SL3(Z) which is Iwahori at p. Suppose f is a noncritical Hecke eigenclass in the homology of Γ with trivial Qp-coefficients. In [4] we outlined a method to compute to any desired degree of accuracy a lift of f to a homology class F with coefficients in a module of p-adic distributions with trivial highest weight. Then we studied how to deform F to an analytic family of homology classes in distribution modules with varying highest p-adic weight. In this paper we explain how to realize these deformations in an actual computer program, and we report on our initial computations of examples. The calculation boils down to row reduction of an infinite matrix over a ring R of power series in three variables over Zp. To carry this out, we must approximate, using finite quotients of R. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Special Issue (9 - 2021), pp. 47-65