Ghost Hecke eigenforms modulo p and the non-semi-simple stratum at the prime 2

Abstract

The space of cusp forms S₂(Γ₀(2)) is zero-dimensional over C. Modulo a prime p, we construct a non-zero formal Hecke eigenform χ_p whose eigenvalues are given by the divisor function τ(m) mod p for m coprime to p, generating a one-dimensional module for a weight-collapsed Hecke algebra. We prove that χ_p is the unique such object among completely multiplicative families, and that it is excluded from the class of automatic sequences in every base b ≥ 2; its generating function satisfies no isolating Mahler equation in any base, and no Mahler equation at all in the base p. The ghost admits a canonical lift, unique up to a unit, to a formal Hecke eigenform modulo p^r for all r ≥ 1, given by the normalised Eisenstein q-expansion with coefficients σ_{p-1}^{(p)}(n). At p = 2 the weight-collapsed algebra coincides with the classical weight-2 Hecke algebra, χ₂ is the mod 2 reduction of the normalised coefficient sequence of the classical eigenform E_{2,2} ∈ M₂(Γ₀(2)), and the reductions of this normalised sequence modulo 2^r form a uniserial Hecke module on which the Hecke algebra acts through the scalar ring Z/2^r—a non-semi-simple stratum, invisible to the semi-simple quotient and to the characteristic-zero dimension formula.