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Isotropic vectors of perfect symmetric bilinear form over Z/p^nZ

Source: Alibaba Global Math Competition 2021, Problem 18

July 4, 2021
vectorlinear algebracollege contestsmodular arithmetic

Problem Statement

Let pp be an odd prime number, and let m≥0m \ge 0 and N≥1N \ge 1 be integers. Let Λ\Lambda be a free Z/pNZ\mathbb{Z}/p^N\mathbb{Z}-module of rank 2m+12m+1, equipped with a perfect symmetric Z/pNZ\mathbb{Z}/p^N\mathbb{Z}-bilinear form ( , ):Λ×Λ→Z/pNZ.(\, ,\,): \Lambda \times \Lambda \to \mathbb{Z}/p^N\mathbb{Z}. Here ``perfect'' means that the induced map \Lambda \to \text{Hom}_{\mathbb{Z}/p^N\mathbb{Z}}(\Lambda, \mathbb{Z}/p^N\mathbb{Z}),   x \mapsto (x,\cdot) is an isomorphism. Find the cardinality of the set {x∈Λ:(x,x)=0},\{x \in \Lambda: (x,x)=0\}, expressed in terms of p,m,Np,m,N.
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