CONGRUENCES FOR HECKE EIGENVALUES VIA PERIOD POLYNOMIALS

Abstract

We study a conjecture motivated by Coleman and Stein’s work on approximating eigenforms of infinite slope by those of finite slopes, which was recast by Rustom [‘Congruences modulo prime powers of Hecke eigenvalues in level 1’, Res. Number Theory 5 (1) (2019), Article no. 10, 27 pages] as a congruence for the second Fourier coefficient. We establish the weight condition required for this coefficient to be divisible by nine and obtain the full conjecture under a natural ramification hypothesis. The proof uses period polynomials, especially Zagier’s description of the Hecke action on them.