Estimates for lattice points of quadratic forms with integral coefficients modulo a prime number square

Let be a nonsingular quadratic form with integer coefficients, n be even. Let V = V Q = V p 2 denote the set of zeros of Q ( x ) in Z p 2 , p be an odd prime, and | V | denote the cardinality of V . In this paper, we are interested in giving an upper bound of the number of integer solutions of the congruence Q ( x ) ≡ 0 ( mod p 2 ) in small boxes of the type { x ∈ Z p 2 n | a i ⩽ x i < a i + m i , 1 ⩽ i ⩽ n } centered about the origin, where a i , m i ∈ Z , and 0 < m i < p 2 for 1 ⩽ i ⩽ n . MSC: 11E04, 11E08, 11E12, 11P21.