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Parity of the partition function in quadratic progressions

This paper proves a 2010 conjecture regarding the parity of the partition function in quadratic progressions by employing a novel geometric approach involving twisted Borcherds products, CM points, and Galois representations to demonstrate that the values contain infinitely many even and odd terms.

Original authors: Ken Ono, Ashvin Swaminathan

Published 2026-08-13

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Ken Ono, Ashvin Swaminathan

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, infinite library where every book is a number, and inside each book, there are countless ways to rearrange its pages. In the world of mathematics, this is the realm of "partitions." A partition is simply a way to break a whole number down into a sum of smaller positive numbers. For instance, the number 4 can be broken down in five different ways: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. Mathematicians call the total count of these ways the "partition function," often written as $p(n)$ .

For over a century, mathematicians have been fascinated by the hidden rhythms in these numbers. They've discovered that if you look at certain patterns, the answers often follow strict rules, like always ending in zero when divided by 5 or 7. But there is one stubborn, chaotic mystery that has resisted all attempts at taming: the "parity" of these numbers. Parity simply means whether a number is even or odd. Does the sequence of partition numbers flip-flop randomly between even and odd, or does it get stuck in a pattern? For a long time, no one could prove if the numbers were truly random or if they secretly favored one side. This paper dives into that chaos, specifically looking at numbers formed by a special quadratic formula, to see if the even and odd values dance together or if one partner leads the whole dance.

The authors of this paper, Ken Ono and Ashvin Swaminathan, have finally solved a specific, long-standing conjecture about this dance. They proved that for a certain family of numbers generated by the formula $Dm^2 + 1$ (where $D$ is a specific type of number and $m$ is an integer), the partition values are not stuck being all even or all odd. Instead, they proved that both even and odd values appear infinitely many times.

To crack this code, the authors didn't just crunch numbers; they built a bridge between two very different worlds of mathematics: the arithmetic of counting (partitions) and the geometry of shapes (elliptic curves). They treated the partition numbers as if they were the coordinates of a map. Using a clever mathematical tool called a "mock theta function" (a type of number pattern discovered by the legendary Srinivasa Ramanujan), they translated the problem of counting partitions into a problem about the shape of a curve.

The key insight was geometric. They imagined the numbers as points on a special map called a "modular curve." They found that the "evenness" or "oddness" of the partition numbers was encoded in the behavior of specific points on this map, known as "CM points." The authors showed that if you look at these points through a special lens (reducing them modulo 2, which is like checking if a number is even or odd), the points remain distinct and don't collapse into each other. This distinctness is crucial. If the points stayed distinct, it meant the partition numbers couldn't all be the same parity; they had to vary.

They used a "geometric argument" to rule out the possibility that all the numbers were even (which would make the map look empty) and the possibility that all were odd (which would make the map look like a smooth, unbroken line). Since the map had "poles" (sharp spikes) that survived the reduction, the numbers had to be a mix. Once they proved that at least one even and at least one odd number existed in the sequence, they used a separate arithmetic rule to show that this single occurrence forces the pattern to repeat forever.

The result is a definitive proof: for these specific quadratic progressions, the partition function is a true coin toss, flipping between even and odd infinitely often. The paper also provides a "speed limit" for this discovery, giving a formula to calculate the smallest number needed to find the first even and first odd values. This work doesn't just solve a puzzle; it introduces a new geometric toolkit that can be used to solve similar riddles about other number sequences, showing that sometimes, to understand the rhythm of numbers, you have to look at the shape of the space they live in.

Technical Summary: Parity of the Partition Function in Quadratic Progressions

Problem Statement
The parity of the partition function $p(n)$ , which counts the number of ways to write a positive integer $n$ as a sum of positive integers, remains a stubbornly open problem in partition theory. While Ramanujan's congruences established regularities modulo 5, 7, and 11, the distribution of $p(n)$ modulo 2 is poorly understood. The Parkin–Shanks conjecture posits that $p(n)$ is even and odd with natural density $1/2$ , but unconditional bounds remain weak.

A specific instance of this difficulty involves restricting the argument of $p(n)$ to quadratic progressions. In 2010, the first author conjectured that for every square-free integer $D > 1$ with $D \equiv 23 \pmod{24}$ , the sequence
$p\left(\frac{Dm^2 + 1}{24}\right)$
contains infinitely many even and infinitely many odd terms as $m$ ranges over positive integers coprime to 6. Prior to this work, essentially nothing was known for polynomials of degree $\ge 2$ .

Methodology and Key Ideas
The authors resolve this conjecture by introducing a geometric framework that connects the arithmetic of the partition function to the geometry of modular curves and CM points. The methodology proceeds through three main stages:

Mock Theta Functions and Borcherds Products:
The authors utilize the fact that Ramanujan's third-order mock theta function $f(q)$ encodes $p(n)$ modulo 4. Specifically, the generating function $P(q) = \sum p(n)q^n$ satisfies $P(q) \equiv f(q) \pmod 4$ . They construct a weight 0 meromorphic modular form $\Psi_D$ for $\Gamma_0(6)$ using the twisted Borcherds lift of Bruinier and the first author. The divisor of $\Psi_D$ consists of two packets of CM points on $X_0(6)$ with discriminant $-D$ . The logarithmic derivative of this product, denoted $\omega_D = \frac{1}{\sqrt{-D}} d \log \Psi_D$ , is a weight 2 meromorphic differential.
Geometric Analysis in Characteristic 2:
The core innovation is the analysis of $\omega_D$ in characteristic 2. The authors establish that:
- The two packets of CM points in the divisor of $\Psi_D$ share a common "2-orientation" (specifically, their order-2 level subgroups correspond to the same prime $\mathfrak{p}_2$ above 2 in the CM field $K = \mathbb{Q}(\sqrt{-D})$ ).
- Due to the rigidity of canonical lifts (Serre–Tate theory), distinct CM points with this common orientation remain distinct after reduction modulo 2.
- The residues of $\omega_D$ at these CM points are 2-adic units. Consequently, the simple poles of $\omega_D$ survive reduction modulo 2 on the ordinary component $C_\infty$ of the special fiber of $X_0(6)$ .
Comparison with Eisenstein Series:
The Fourier expansion of $\omega_D$ at the cusp $\infty$ is a Lambert series encoding the parities of the partition values in question. The authors compare this expansion to a holomorphic weight 2 Eisenstein series $E_D$ constructed to match the "all odd" scenario.
- If all partition values were even, the reduction of $\omega_D$ would vanish identically, contradicting the survival of the CM poles.
- If all partition values were odd, the reduction of $\omega_D$ would be congruent to the reduction of $E_D$ . However, $E_D$ is holomorphic (regular) at the CM points, whereas the reduction of $\omega_D$ retains simple poles there. This contradiction rules out constant odd parity.

Key Results
The paper proves the following:

Theorem 1: For every square-free $D > 1$ $D > 1$ with $D \equiv 23 \pmod{24}$ $D \equiv 23 (mod 24)$ , the values $p\left(\frac{Dm^2 + 1}{24}\right)$ $p (\frac{D m ^{2} + 1}{24})$ for $(m, 6)=1$ $(m, 6) = 1$ include infinitely many even and infinitely many odd terms.
- Furthermore, the paper provides effective bounds for the first occurrence of each parity in terms of the class number $h(-D)$ :
  $m_{\text{odd}} \le 12h(-D) + 2$
  $m_{\text{even}} \le (12h(-D) + 2) \prod_{\ell|D} (\ell + 1)$
Theorem 2: A generalized nonconstancy criterion is established for sequences arising from generalized twisted Borcherds products. This theorem isolates the necessary geometric inputs (integrality of the differential, existence of simple CM poles with unit residues, and the existence of a matching holomorphic modular form) required to prove that a sequence is not constant modulo 2.

Significance and Claims
The paper claims to solve a specific, long-standing conjecture regarding the parity of the partition function in quadratic progressions, a case where previous methods had yielded no results. The significance lies in the "geometric" nature of the proof:

It recasts a combinatorial problem about partition coefficients into a problem about the behavior of poles of modular differentials in characteristic 2.
It demonstrates that the "survival" of CM poles under reduction modulo 2, combined with the uniqueness of canonical lifts, provides a robust mechanism to rule out constant parity patterns.
The authors note that while the geometric argument (Theorem 2) establishes nonconstancy (i.e., ruling out "all even" or "all odd"), the step from "one occurrence" to "infinitely many" relies on separate arithmetic propagation results involving Hecke algebras and Galois representations (specifically, the local nilpotence of certain Hecke operators). Thus, the paper does not claim that the geometric method alone yields infinitude for general Borcherds data, but rather that it provides the crucial nonconstancy step which, when combined with known arithmetic propagation theorems, yields the full result.

The algebraic identities underpinning the proof were formalized and verified in Lean by AxiomProver, ensuring the correctness of the intricate combinatorial and modular form calculations.

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