Lucas sequences and quadratic orders

We consider the Lucas sequences (Un)n ≥ 0 defined by U0 = 0, U1 = 1, and Un =  PUn–1 – QUn–2 for non-zero integral parameters P, Q such that Δ = P2 – 4Q is not a square. We use the arithmetic of the quadratic order with discriminant Δ to investigate the zeros and the period length of the sequence (Un)n ≥ 0 modulo a positive integer d coprime to Q. For a prime p not dividing Q, we give precise formulas for p-powers, we determine the p-adic value of Un, and we connect the results with class number relations for quadratic orders.