engleski

Diophantine m-tuples in finite fields and modular forms

For a prime p, a Diophantine m-tuple in finite field 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 $N^{; ; ; ; (m)}; ; ; ; (p)=\ds \frac{; ; ; ; 1}; ; ; ; {; ; ; ; 2^{; ; ; ; m \choose 2 }; ; ; ; }; ; ; ; \frac{; ; ; ; p^m}; ; ; ; {; ; ; ; m!}; ; ; ; + o(p^m)$, and also show that if p>2^{; ; ; ; 2m-2}; ; ; ; m^2, then there is at least one Diophantine m-tuple in F_p.

Diophantine m-tuples, modular forms, finite fields, elliptic curves

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