Number theoretic properties of Wronskians of Andrews-Gordon series

Abstract

For positive integers $1\le i\le k$, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon $q$-series $$\prod _{1\le n\not\equiv 0,\pm i(\mathrm{mod}2k+1)}\frac{1}{1-q^n}.$$ 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 $0,\pm a(\mathrm{mod}b)$, 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 $p\ge 5$ is prime, and the quotient is non-zero, then it is essentially the locus of characteristic $p$ supersingular $j$-invariants in characteristic $p$.

13 pages, AMS-LaTeX