Number theoretic properties of Wronskians of Andrews-Gordon series

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 $p$.