Number theoretic properties of Wronskians of Andrews-Gordon series

TL;DR

This paper investigates the arithmetic properties of Wronskians derived from Andrews-Gordon series, revealing their connections to partition identities, conformal field theory, and supersingular elliptic curves.

Contribution

It determines the vanishing conditions of these Wronskians, uncovers new partition identities, and links the quotients to supersingular elliptic curves in characteristic p.

Findings

Identified when Wronskians vanish, leading to new partition identities.

Established a connection between Wronskians and supersingular elliptic curves.

Derived explicit formulas relating partition counts for specific moduli.

Abstract

For positive integers $1\leq i\leq k$, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon $q$-series $$ \prod_{1\leq n\not \equiv 0,\pm i\pmod{2k+1}}\frac{1}{1-q^n}. $$ This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of $(2,2k+1)$ Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if $P_{b}(a;n)$ denotes the number of partitions of $n$ into parts which are not congruent to $0, \pm a\pmod b$, then for every positive integer $n$ we have $$ P_{27}(12; n)=P_{27}(6;n-1) + P_{27}(3;n-2). $$ We also show that these quotients classify supersingular elliptic curves in characteristic $p$. More precisely, if $2k+1=p$, where $p\geq 5$ is prime, and the quotient is non-zero, then it is essentially the locus of characteristic $p$ supersingular $j$-invariants in characteristic $p$.