Diophantine m-tuples in finite fields and modular forms

Abstract

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 $$N^{(m)}(p)=\displaystyle \frac{1}{2^{m \atopwithdelims ()2 }}\frac{p^m}{m!} + o(p^m)$$ , and also show that if $$p>2^{2m-2}m^2$$ , then there is at least one Diophantine m-tuple in $$\mathbb {F}_p$$ .