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 $X_0(6)$. 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 $G_\ell$ of weight $\ell+1$ on $Γ_0(\ell)$, compute their Fricke transformations, and prove the exact identities $$ \mathcal T_5(G_5)=-5E_6,\qquad \mathcal T_7(G_7)=-7E_8,\qquad \mathcal T_{13}(G_{13})=13E_{14}. $$ 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.