Diophantine m-tuples in finite fields and modular forms

For a prime p, a Diophantine m-tuple in ${\mathbb{F}}_p$ is a set of m nonzero elements of ${\mathbb{F}}_p$ 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 ${\mathbb{F}}_p$ 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 $N^{(m)}(p)=\frac{1}{2^{m \choose 2 }}\frac{p^m}{m!} + o(p^m)$,andalsoshowthatif$p>2^{2m-2}m^2$, then there is at least one Diophantine m-tuple in ${\mathbb{F}}_p$.