The partition function and elliptic curves

Abstract

For each $n\geq 1$, we express the partition function $p(n)$ as a CM trace on $X_0(6)$ of the discriminant $\Delta_n:=1-24n$ invariants of a weight 0 weak Maass function $P$ that records where CM elliptic curves sit on $X(1)$, together with their canonical first-order "CM tangent'', the diagonal local slope of the CM isogeny relation on $X(1)\times X(1)$. In this viewpoint, we obtain a formula for $p(n)\!\!\pmod{\ell},$ when $\ell$ is inert in $\mathbb{Q}(\sqrt{\Delta_n}),$ as a Brandt-module pairing $\langle u_{\Delta_n},v_P\rangle$ that is assembled from oriented optimal embeddings of Eichler orders. For $\ell \in \{5, 7, 11\}$ and $j\geq 1$, we obtain a new proof of the Ramanujan congruences $$ p(5^j n +\beta_5(j))\equiv 0\pmod{5^j}, $$ $$ p(7^j n +\beta_7(j))\equiv 0\pmod{7^{ [ j/2]+1}}, $$ $$ p(11^jn+\beta_{11}(j))\equiv 0\pmod{11^j}, $$ where $\beta_m(j)$ is the unique residue $0\le \beta<m^j$ with $24\,\beta_m(j)\equiv 1\pmod{m^j}$. The key point is a "bonus valuation" that stems from the fact that the supersingular locus of $X_0(6)_{\mathbb{F}_{\ell}}$ lies over $\{0, 1728\}$ for $\ell \in \{5, 7, 11\}.$ This special property, combined with the uniform growth of the $\lambda$-adic valuations of the number of oriented optimal embeddings, explains these congruences. More generally, we give a portable genus 0 template showing that the Watson--Atkin $U_\ell$-contraction works uniformly for suitable traces of singular moduli for genus 0 modular curves with $\ell\nmid N.$