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, $\operatorname{spt}(n)$, are $\ell= 5, 7, 13$. 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.