The smallest-parts function and elliptic curves
Abstract
Recently, Ono established a new connection between partition congruences and the arithmetic of elliptic curves by expressing the partition function p(n) as a CM trace on the modular curve X0(6). This approach yields new proofs of Ramanujan-type congruences for the partition function modulo powers of $5$, $7$, and $11$, revealing how the geometry of supersingular elliptic curves and the arithmetic of oriented optimal embeddings govern these congruence properties. Motivated by Ono's work, we develop an analogous passage from CM traces to congruences modulo $5$, $7$, and $13$ for the for Andrews's smallest-parts partition function spt(n), employing the established theta-lift and modular-trace methods suited to this setting. With P the Bruinier--Ono weight-zero weak Maass function and f an explicit weakly holomorphic modular function on Γ0(6), the Ahlgren--Andersen formula expresses spt(n) as one twelfth of the discriminant 1−24n CM trace of f−P. We explain the theta-lift normalization, the contribution of the two orientations, and the harmonic-Maass-form uniqueness argument. Following Garvan's work, we construct integral forms Gℓ of weight ℓ+1 on Γ0(ℓ), compute their Fricke transformations, and prove the exact identities 𝒯5(G5)=−5E6, 𝒯7(G7)=−7E8, 𝒯13(G13)=13E14. Their reductions yield the three congruences on the Andrews's smallest-parts partition function. We also give a supersingular specialization of the trace of f, with actual reduction multiplicities, and explain the role of imprimitive forms, the conductor restriction in CM reduction, and the distinction between the primes $11$ and $13$. These results have been verified in Lean using AxiomProver.
Related
- Ramanujan-type congruences for the smallest parts function
- Class polynomial obstructions to canonical congruences for the smallest parts function
- Frey-Mazur theorem for semistable elliptic curves over function fields
- Tamagawa numbers and torsion of elliptic curves over function fields
- Rank stability of elliptic curves over prime cyclic extensions of function fields
BibTeX
@misc{singh2026smallestparts,
title = {{The smallest-parts function and elliptic curves}},
author = {Ajit Singh},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04387},
eprint = {2611.04387},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04387},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The smallest-parts function and elliptic curves
AU - Ajit Singh
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04387
UR - https://arxiv.org/abs/2611.04387
AB - Recently, Ono established a new connection between partition congruences and the arithmetic of elliptic curves by expressing the partition function p(n) as a CM trace on the modular curve X_0(6). This approach yields new proofs of Ramanujan-type congruences for the partition function modulo powers of $5$, $7$, and $11$, revealing how the geometry of supersingular elliptic curves and the arithmetic of oriented optimal embeddings govern these congruence properties. Motivated by Ono's work, we develop an analogous passage from CM traces to congruences modulo $5$, $7$, and $13$ for the for Andrews's smallest-parts partition function spt(n), employing the established theta-lift and modular-trace methods suited to this setting. With P the Bruinier--Ono weight-zero weak Maass function and f an explicit weakly holomorphic modular function on Γ_0(6), the Ahlgren--Andersen formula expresses spt(n) as one twelfth of the discriminant 1−24n CM trace of f−P. We explain the theta-lift normalization, the contribution of the two orientations, and the harmonic-Maass-form uniqueness argument. Following Garvan's work, we construct integral forms G_ℓ of weight ℓ+1 on Γ_0(ℓ), compute their Fricke transformations, and prove the exact identities 𝒯_5(G_5)=−5E_6, 𝒯_7(G_7)=−7E_8, 𝒯_13(G_13)=13E_14. Their reductions yield the three congruences on the Andrews's smallest-parts partition function. We also give a supersingular specialization of the trace of f, with actual reduction multiplicities, and explain the role of imprimitive forms, the conductor restriction in CM reduction, and the distinction between the primes $11$ and $13$. These results have been verified in Lean using AxiomProver.
ER -