Parity of the partition function in quadratic progressions

Abstract

The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If $1<D\equiv 23\pmod{24}$ is square-free and only divisible by primes $\ell\equiv 1, 7\pmod 8$, then both parities occur infinitely often among $$ p\left(\frac{Dm^2+1}{24}\right), $$ with $(m,6)=1.$ The argument takes place on the modular curve $X_0(6)$ and shows that parity along these thin orbits is \emph{not constant}. The proof connects classical identities for the partition generating function, through the method of (twisted) Borcherds products, to the arithmetic geometry of {\it ordinary} CM fibers on the Deligne-Rapoport model of $X_0(6)$ in characteristic 2. This result is a special case of a general theorem for the coefficients of suitable vector-valued weight 1/2 harmonic Maass forms that satisfy a "Heegner packet'' condition.