Scarcity of congruences for the partition function

TL;DR

This paper investigates the rarity of certain congruences related to the partition function, showing that if they exist for specific cases, they are extremely uncommon, using advanced modular forms and number theory techniques.

Contribution

It proves that congruences of the form $p( ext{linear function of } n) ot\equiv 0 mod ext{prime}$ for $m=1,2$ are exceedingly rare, filling a gap in the understanding of partition congruences.

Findings

Such congruences are extremely scarce if they exist.

The methods involve modular forms of half integral weight and Galois representations.

Uses advanced tools like Radu's work and the arithmetic large sieve.

Abstract

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the primes $\ell=5, 7, 11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0\pmod\ell$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.