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Some congruences for binomial coefficients. II. (English) Zbl 0971.11046

Let \(p= tn+r\) be a prime where \(t\), \(n\), \(r\) are positive integers \((1\leq r\leq t-1)\) and either (1) \(t= k>3\) for a prime \(k\equiv 3\pmod 4\) or (2) \(t=4k\) for a prime \(k\equiv 1\pmod 4\). Let \(H\) be the subgroup with the operator \(r\) of \((\mathbb{Z}/t\mathbb{Z})^\times\) isomorphic to \(\text{Gal} (\mathbb{Q}(\xi_t)/ \mathbb{Q}(\sqrt{-t}))\), \(h\) be the class number of \(\mathbb{Q}(\sqrt{-t})\), \(s= \frac{\varphi(t)} {2}\) and \(d= ((p^s-1)/t) (t-1)= \sum_{j=0}^{s-1} d_j p^j\) \((0\leq d_j< p)\). The existence of integers \(a\), \(b\) was shown by the authors [Proc. Class Field Theory – its Centenary and Prospect (1998), Adv. Stud. Pure Math. 30, 445-461 (2001)] such that \(4p^h= a^2+ tb^2\) and \[ a\equiv\pm \prod_{j=0}^{s-1} (d_j)!\pmod p. \] If we consider \(k=1\) and \(r=1\), then we get Gauss’ statement \(a\equiv \binom {2n}{n}\pmod p\) \((4p= a^2+ 4b^2\), \(\frac{a}{2}\equiv 1\pmod 4)\).
The main result of this paper determines the sign of \(a\) in the following sense: In case (1) \(a\equiv 2p^\beta\pmod t\) and \(a\equiv (-1)^\beta \prod_{j=0}^{s-1} (d_j)!\pmod p\) and in case (2) \(a\equiv -p^\beta\pmod k\) and \(2a\equiv (-1)^\alpha \prod_{j=0}^{k-2} (d_j)!\pmod p\).
The integer \(\beta\) is defined by \[ \beta= \Bigl( \sum_{1\leq i\leq t, -i\in H} i\Bigr)/t \] and \(\alpha= s-\beta\). To prove these results some equalities on Eisenstein sums are presented.

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11T24 Other character sums and Gauss sums
11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
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