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 $N^{(m)}(p)=\frac{1}{2^{\left(\genfrac{}{}{0ex}{}{m}{2}\right)}}\frac{p^m}{m!}+o(p^m)$, and also show that if $p>2^{2m-2}{m}^2$, then there is at least one Diophantine m-tuple in ${\mathbb{F}}_p$.

25 pages; statement of Theorem 10. corrected

References in corpus (1)

• More on Diophantine sextuples

Cited by in corpus (5)

• D(n)-quintuples with square elements
• Rational $D(q)$-quadruples
• Explicit constructions of Diophantine tuples over finite fields
• $F$-Diophantine sets over finite fields
• Rational $D(q)$-quintuples