Class polynomial obstructions to canonical congruences for the smallest parts function
Abstract
We resolve an open conjecture by proving that the only "canonical'' Ramanujan congruence primes for the Andrews' smallest parts function, spt(n), are ℓ= 5, 7, 13. Building on work of Ahlgren and Kim, we show that any such congruence forces square-index coefficients of certain weakly holomorphic forms of weight 1/2 to vanish modulo the congruence prime. A normalized Borcherds product converts this into the vanishing of the derivative of a quotient of two oriented CM polynomials on X0(6). A quaternionic volume estimate in an Eichler order of level $6$ prevents that vanishing when a suitable inert discriminant is small. Treviño's character estimate supplies such a discriminant for every prime at least 1011. The remaining range is covered by the previous calculations of Ahlgren and Kim. These results have been verified in Lean by AxiomProver.
Related
- Ramanujan-type congruences for the smallest parts function
- Calogero-Moser Polynomial Approximations and Dwork Congruences for Complete Flags
- Nonexistence of Constant Congruences for the Cumulative Partition Function
- Four-square polynomial triples: quadratic classification and equal-degree obstructions
- Congruences for the Andrews--Uncu partition function ℰ𝒪u(n)
BibTeX
@misc{rolen2026class,
title = {{Class polynomial obstructions to canonical congruences for the smallest parts function}},
author = {Larry Rolen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27145},
eprint = {2610.27145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27145},
}
RIS
TY - UNPB
TI - Class polynomial obstructions to canonical congruences for the smallest parts function
AU - Larry Rolen
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27145
UR - https://arxiv.org/abs/2610.27145
AB - We resolve an open conjecture by proving that the only "canonical'' Ramanujan congruence primes for the Andrews' smallest parts function, spt(n), are ℓ= 5, 7, 13. Building on work of Ahlgren and Kim, we show that any such congruence forces square-index coefficients of certain weakly holomorphic forms of weight 1/2 to vanish modulo the congruence prime. A normalized Borcherds product converts this into the vanishing of the derivative of a quotient of two oriented CM polynomials on X_0(6). A quaternionic volume estimate in an Eichler order of level $6$ prevents that vanishing when a suitable inert discriminant is small. Treviño's character estimate supplies such a discriminant for every prime at least 10^11. The remaining range is covered by the previous calculations of Ahlgren and Kim. These results have been verified in Lean by AxiomProver.
ER -