The partition function and elliptic curves

TL;DR

This paper links the partition function to CM traces on modular curves, providing new proofs of Ramanujan congruences via elliptic curve invariants and local slope analysis, revealing deep connections between partition theory and algebraic geometry.

Contribution

It introduces a novel CM trace formula for the partition function and offers a new proof of Ramanujan congruences using elliptic curve invariants and modular curve properties.

Findings

Expresses p(n) as a CM trace on X_0(6) involving elliptic curve invariants.

Provides a new proof of Ramanujan congruences for 5, 7, and 11.

Identifies a special property of supersingular loci that explains these congruences.

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 $\ell\nmid N.$