The partition function and elliptic curves

Abstract: We recast congruences for the partition function through the geometry of supersingular elliptic curves. For primes $\ell\ge5$ inert in $K_n=\mathbb{Q}(\sqrt{1-24n})$ (equivalently $\big(\tfrac{1-24n}{\ell}\big)=-1$), we show that $p(n)\pmod{\ell}$ is a "supersingular trace'' on $X_0(6)$: a computable dot product on the level-$6$ Brandt module that is a twisted analogue of the Deuring--Eichler formula for class numbers. In the ramified case $\ell\in{5,7,11}$ (i.e. $\ell\mid(1-24n)$), a geometric "bonus valuation'' along supersingular fibers forces Ramanujan's congruences [ p(5n+4)\equiv0\pmod5,\qquad p(7n+5)\equiv0\pmod7,\qquad p(11n+6)\equiv0\pmod{11}. ] This phenomenon occurs because the supersingular $j$-invariants on $X_0(6)<em>{\mathbb{F}</em>\ell}$ for these primes lie over ${0,1728}$, allowing a certain Serre-Tate pullback to be regular at every supersingular point.