The partition function modulo prime powers

Abstract

Let ≥ 5 be prime, let m ≥ 1 be an integer, and let p(n) denote the partition function. Folsom, Kent, and Ono recently proved that there exists a positive integer b (m) of size roughly m 2 such that the module formed from the Z/ m Z-span of generating functions for p b n+1 24 with odd b ≥ b (m) has finite rank. The same result holds with "odd" b replaced by "even" b. Furthermore, they proved an upper bound on the ranks of these modules. This upper bound is independent of m; it is +12 24. In this paper, we prove, with a mild condition on , that b (m) ≤ 2m − 1. Our bound is sharp in all computed cases with ≥ 29. To deduce it, we prove structure theorems for the relevant Z/ m Z-modules of modular forms. This work sheds further light on a question of Mazur posed to Folsom, Kent, and Ono.

$$\ell $$ ℓ -Adic properties of partition functions

“…Here, we give numerical examples of the main theorems in this paper. In all cases, using the methods in §6 of [6], we find that d (r) and d (spt) are zero.…”

“…Proof. The proof is similar to that of Lemma 3.1 in [6], so we will omit some details and instead note the differences. For (1), we let g j denote the forms in M k (spt,j) congruent to Ψ modulo j , and for (2), we let g j denote the forms in M j−1 ( −1) congruent to Ψ modulo and modulo j , respectively.…”

“…Note that the extra factor of in (3.12), not present in [6], comes from combining the matrices after the slash operator. The rest of the proof is the same: for k = 0 we decompose the matrix γ as…”

“…To state our results, we define a sequence of generating functions related to p r (n) which are analogous to those that appear in [8] and [6],…”

“…Our main theorem is that these functions, reduced modulo powers of , reside in a relatively small space. Fixing integers b ≥ 0, m ≥ 1, following [6] we define (1 − q n ) is Dedekind's eta-function. In Section 4 we will define d (r) and d (r), which are related to the dimension of the kernel of an important operator.…”