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

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