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Binomial coefficients and Jacobi sums. (English) Zbl 0551.12015

In 1828, Gauss proved that if \(p=4n+1\) is a prime, so that \(p=a^ 2+b^ 2\), \(a\equiv 1 (4)\) unique, then \(\left( \begin{matrix} 2f\\ f\end{matrix} \right)\equiv 2a (mod p).\) Many authors have derived similar congruences for various binomial coefficients \(\left( \begin{matrix} rf\\ sf\end{matrix} \right)\), \(1\leq s<r\leq e-1\), where \(p-1=ef,\) in terms of the parameters of the corresponding quadratic partition of p. All these results are ad hoc and the main contents of the paper is to systematize these scattered isolated curiosities into one body, giving all the old results as well as a large number of new ones. This is achieved by using a result which connects the Jacobi sums and these binomial coefficients viz. \(\left( \begin{matrix} rf\\ sf\end{matrix} \right)\equiv (-1)^{sf+1} J_ e(r,e-s) (mod P)\) where P is a factor of p in \({\mathbb{Q}}\) \((e^{2\pi i/p})\). This result was first proved by Whiteman in a slightly different form.
As soon as \(\left( \begin{matrix} rf\\ sf\end{matrix} \right)\) is coupled (mod p) with a Jacobi sum, it is coupled (mod p) with cyclotomic numbers for the factorization \(p-1=ef,\) which, if known in terms of the parameters of the quadratic partition of p, then gives a congruence of the required type. The authors, however, stick to the Jacobi sums without any mention of the cyclotomic numbers (i,j).
In addition, some interesting relations amongst the \(\left( \begin{matrix} rf\\ sf\end{matrix} \right)\) are derived via the quantity \(n^{(p-1)/m}\), which is an mth root of unity mod p and for which expressions are known in some cases, in terms of the parameters of the quadratic partition of p. - The paper opens up many avenues for further research.
Reviewer: A.R.Rajwade

MSC:

11T22 Cyclotomy
11A15 Power residues, reciprocity
11E16 General binary quadratic forms
05A10 Factorials, binomial coefficients, combinatorial functions
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