The partition function and elliptic curves

Abstract

The Bruinier-Ono formula expresses the partition number $p(n)$ as a trace of `non-holomorphic' singular moduli of discriminant $Δ_n:=1-24n$ CM points on $X_0(6).$ We interpret this trace through the geometry of CM points. Each nonholomorphic contribution is the value of the weight-two completion $E_2^*$ at a CM point, which is a canonical invariant of the underlying elliptic curve, determined by the diagonal `tangent' of the CM isogeny relation. This turns the trace into a quantity that can be reduced to the supersingular locus that is organized by Deuring-Eichler multiplicities and a Brandt-module pairing. For primes $ \ell\geq 5$ that are nonsplit in $\mathbb{Q}(\sqrt{Δ_n})$, we obtain a supersingular trace formula on $X_0(6)$ over $\overline{\mathbb{F}}_{\ell}$. For the special primes $\ell=5,7,11$, this sheds new light on Ramanujan's classical partition congruences. These primes are special because they are the only ones for which the supersingular locus of $X_0(6)$ lies over $j\in \{0, 1728\}.$ This perspective offers a moduli-theoretic framework for Ramanujan's congruences modulo powers of these primes, organized through elliptic curves. The two new algebraic identities at the heart of this framework, as opposed to the classical results it builds on, were formalized and verified in Lean by AxiomProver.