Probabilistic enumerative geometry over p-adic numbers: linear spaces on complete intersections

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We compute the expectation of the number of linear spaces on a complete intersection in $p$-adic projective space. Here random means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a complete intersection tends to $1$. In the case of the number of lines on a cubic in three-space and on the intersection of two quadrics in four-space, we give an explicit formula for this expectation.