Parity of the partition function in quadratic progressions

Abstract

The parity of the partition function $p(n)$ is one of the most stubborn problems in partition theory. In 2010, the first author conjectured that, for every square-free $1<D\equiv 23(\mathrm{mod}24)$, the values \[ p\!\left(\frac{Dm^2+1}{24}\right), \] as $m$ ranges over the positive integers with $(m,6)=1$, include infinitely many even and infinitely many odd terms. We prove this conjecture. The key new idea is geometric. Logarithmic derivatives of twisted Borcherds products built from Ramanujan's third-order mock theta functions recast the problem in terms of CM points of discriminant $-D$ on $X_0(6)$. The uniqueness of canonical lifts from characteristic $2$ to characteristic zero shows that the CM points supporting the poles remain distinct after reduction modulo $2$. This fact, combined with a comparison with Eisenstein series, rules out both constant parity patterns. A Galois representation argument then gives infinitely many values of each parity. More generally, the method applies to analogous sequences arising from suitable generalized twisted Borcherds products. The algebraic identities at the heart of this paper were formalized and verified in Lean by AxiomProver.

Main theorem strengthened to cover all fund disc -D=1 mod 24. Ashvin Swaminathan added as a co-author, and all identities now verified in Lean