Diophantine m-tuples in finite fields and modular forms

TL;DR

This paper derives formulas for counting Diophantine m-tuples in finite fields, revealing connections to modular forms and establishing asymptotic behavior and existence results for such tuples.

Contribution

It provides explicit formulas for the number of Diophantine m-tuples in finite fields for m=2,3,4, and links these counts to modular form Fourier coefficients.

Findings

Formulas for N^{(m)}(p) for m=2,3,4

Asymptotic estimate for N^{(m)}(p) as p grows

Existence of Diophantine m-tuples in large finite fields

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