Generalisation of a congruence of Gauss
For a prime \(p\equiv 1(4)\), \(p=a^2+b^2\), \(a\equiv 1(4)\) unique, Gauss proved that \(\binom{(p-1)/2}{(p-1)/4}\equiv 2a\pmod p\). An improvement of this congruence is \(\binom{(p-1)/2}{(p-1)/4}\equiv (-4)^{(p-1)/4}(a+bi)\pmod{p^2}\), where \(a+bi\in\mathbb Z_p\) and \(| a+bi|_p=1\). This was proved by \textit{S. Chowla}, \textit{B. Dwork} and \textit{R. Evans} [J. Number Theory 24, 188--196 (1986; Zbl 0596.10003)]. Here the author generalizes this congruence to \[ \binom{(mp^{r-1})/2}{(mp^{r-1})/4}\equiv \binom{(mp^{r-1}-1)/2}{(mp^{r-1}-1)/4}(-4)^{\{mp^{r-1}\cdot (p-1)/4\}}\cdot (a+bi)\pmod {p^{2r}} \] where \(m\equiv 1(4)\). This congruence and many others are proved using Gauss and Jacobi sums and the theory of the \(p\)-adic \(\Gamma\)-function. The author remarks that these congruences are very easy to prove modulo \(p^r\) by the use of the theory of elliptic curves or the theory of formal group laws.
scientific article; zbMATH DE number 4061313
scientific article