Diophantine m-tuples in finite fields and modular forms

For a prime p, a Diophantine m-tuple in F-p is a set of m nonzero elements of F-p with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for the number N-(m)(p) of Diophantine m-tuples in F-p for m = 2,3 and 4. Fourier coefficients of certain modular forms appear in the formula for the number of Diophantine quadruples. We prove that asymptotically N-(m)(p) = 1/2((m)(2))p(m)/m! + o(p(m)), and also show that if p > 2(2m-2)m(2), then there is at least one Diophantine m-tuple in F-p.