Binomial coefficients and Jacobi sums

Abstract

Throughout this paper e e denotes an integer ⩾ 3 \geqslant 3 and p p a prime ≡ 1 ( mod e ) \equiv \;1\ \pmod e . With f f defined by p = e f + 1 p = ef + 1 and for integers r r and s s satisfying 1 ⩽ s > r ⩽ e − 1 1 \leqslant s > r \leqslant e - 1 , certain binomial coefficients ( r f s f ) \left ( {\begin {array}{*{20}{c}} {rf} \\ {sf} \\ \end {array} } \right ) have been determined in terms of the parameters in various binary and quaternary quadratic forms by, for example, Gauss [13], Jacobi [19, 20], Stern [37-40], Lehmer [23] and Whiteman [42, 45, 46]. In § 2 \S 2 we determine for each e e the exact number of binomial coefficients ( r f s f ) \left ( {\begin {array}{*{20}{c}} {rf} \\ {sf} \\ \end {array} } \right ) not trivially congruent to one another by elementary properties of number theory and call these representative binomial coefficients. A representative binomial coefficient is said to be of order e e if and only if ( r , s ) = 1 (r,s) = 1 . In § § 3 − 4 \S \S 3-4 , we show how the Davenport-Hasse relation [7], in a form given by Yamamoto [50], leads to determinations of n ( p − 1 ) / m {n^{(p - 1)/m}} in terms of binomial coefficients modulo p = e f + 1 = m n f + 1 p = ef + 1 = mnf + 1 . These results are of some interest in themselves and are used extensively in later sections of the paper. Making use of Theorem 5.1 relating Jacobi sums and binomial coefficients, which was first obtained in a slightly different form by Whiteman [45], we systematically investigate in § § 6 − 21 \S \S 6-21 all representative binomial coefficients of orders e = 3 , 4 , 6 , 7 , 8 , 9 , 11 , 12 , 14 , 15 , 16 , 20 e = 3,4,6,7,8,9,11,12,14,15,16,20 and 24 24 , which we are able to determine explicitly in terms of the parameters in well-known binary quadratic forms, and all representative binomial coefficients of orders e = 5 , 10 , 13 , 15 , 16 e = 5,10,13,15,16 and 20 20 , which we are able to explicitly determine in terms of quaternary quadratic decompositions of 16 p 16p given by Dickson [9], Zee [51] and Guidici, Muskat and Robinson [14]. Some of these results have been obtained by previous authors and many new ones are included. For e = 7 e = 7 and 14 14 we are unable to explicitly determine representative binomial coefficients in terms of the six variable quadratic decomposition of 72 p 72p given by Dickson [9] for reasons given in § 10 \S 10 , but we are able to express these binomial coefficients in terms of the parameter x 1 {x_1} in this system in analogy to a recent result of Rajwade [34]. Finally, although a relatively rare occurrence for small e e , it is possible for representative binomial coefficients of order e e to be congruent to one another ( mod p ) \pmod p . Representative binomial coefficients which are congruent to ± 1 \pm 1 times at least one other representative for all p = e f + 1 p = ef + 1 are called Cauchy-Whiteman type binomial coefficients for reasons given in [17] and § 21 \S 21 . All congruences between such binomial coefficients are carefully examined and proved (with the sign ambiguity removed in each case) for all values of e e considered. When e = 24 e = 24 there are 48 48 representative binomial coefficients, including those of lower order, and it is shown in § 21 \S 21 that an astonishing 43 43 of these are Cauchy-Whiteman type binomial coefficients. It is of particular interest that the sign ambiguity in many of these congruences does not arise from any expression of the form n ( p − 1 ) / m {n^{(p - 1)/m}} in contrast to the case for all e > 24 e > 24 .