Number Theoretic Properties of Wronskians of Andrews-gordon Series

For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon q-series ∏ 1≤n6≡0,±i (mod 2k+1) 1 1− qn . 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 Pb(a;n) denotes the number of partitions of n into parts which are not congruent to 0,±a (mod b), then for every positive integer n we have P27(12;n) = P27(6;n − 1) + P27(3;n − 2). We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k +1 = p, where p ≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of characteristic p supersingular j-invariants in characteristic p.