Parity of the partition function in quadratic progressions
Ken Ono
TL;DR
The paper establishes that for square-free $D$ with $1<D\equiv 23\pmod{24}$ and primes dividing $D$ congruent to $1$ or $7$ mod $8$, both parities occur infinitely often among $p\left(\frac{D m^2+1}{24}\right)$ with $(m,6)=1$. It achieves this by constructing twisted Borcherds products on the modular curve $X_0(6)$ from Ramanujan’s mock theta functions, whose CM divisors encode parity information, and by analyzing ordinary CM fibers modulo $2$ via Serre–Tate theory on the Deligne–Rapoport model. A global Lambert-series identity links $d\log\Psi_D$ to the parity data of $p(n)$, while a characteristic-$2$ kernel result ensures nontrivial residues imply non-squareness, yielding infinite occurrences of both parities. The approach generalizes to coefficients of suitable vector-valued weight $\tfrac12$ harmonic Maass forms with Heegner-packet structure, linking partition parity to arithmetic geometry and class-field theory. Overall, the paper provides a novel, unconditional route to nontrivial parity results for non-linear quadratic progressions in the partition function.
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.