Lucas sequences and quadratic orders

We consider the Lucas sequences (U _n ) _{n ≥ 0} defined by U ₀ = 0, U ₁ = 1, and U _n =  PU _n–1 – QU _n–2 for non-zero integral parameters P, Q such that Δ = P ² – 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 (U _n ) _{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 U _n , and we connect the results with class number relations for quadratic orders.