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

Let [InlineEquation not available: see fulltext.] be a nonsingular quadratic form with integer coefficients, n be even. Let [InlineEquation not available: see fulltext.] denote the set of zeros of [InlineEquation not available: see fulltext.] in [InlineEquation not available: see fulltext.], p be an odd prime, and [InlineEquation not available: see fulltext.] 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 [InlineEquation not available: see fulltext.] in small boxes of the type [InlineEquation not available: see fulltext.] centered about the origin, where [InlineEquation not available: see fulltext.], and [InlineEquation not available: see fulltext.] for [InlineEquation not available: see fulltext.]. MSC:11E04, 11E08, 11E12, 11P21. © 2014, Hakami; licensee Springer.