The BP<n> cohomology of elementary abelian groups

In this paper we study E^*BV_k, where E=BP<m,n> is a cohomology theory with coefficient ring F_p[v_m,...,v_n] (if m>0) or Z_(p)[v_1,...,v_n] (if m=0). We use ideas from the theory of multiple level structures, developed in earlier work of the author with John Greenlees. Our results apply when k is less than or equal to w=n+1-m. If k<w we find that E^*BV_k has no v_m-torsion. When k=w, we show that the v_m-torsion is annihilated by the ideal I_{n+1}=(v_m,...,v_n), and that it is a free module on one generator over the ring F_p[[x_0,...,x_{w-1}]]. We give three very different formulae for this generator; it is not at all obvious that these give the same element, and we only have a rather indirect proof of this.