Gauss sums and multinomial coefficients
Suppose that \(p=tn+r\) is a prime which splits completely in the imaginary quadratic field \(\mathbb Q[\sqrt{-t}]\). \textit{L. Stickelberger} [Math. Ann. 37, 321--367 (1890; JFM 22.0100.01)] proved that if \(t\notin \{3,4,8\}\) and \(-t\) is a fundamental discriminant, then there are integers \(a,b\) such that \(4p^h=a^2+tb^2\), where \(h\) is the class number of \(\mathbb Q[\sqrt{-t}]\)~. A classical problem is to characterize \(a\) by congruence conditions. In the paper under review, the author shows how, since \(\mathbb Q[\sqrt{-t}]\) may be identified with a subfield of the \(p\)-adic field \(\mathbb Q_p\) whenever \(p\) splits in \(\mathbb Q[\sqrt{-t}]\), congruences expressing \(a\) modulo \(p\) may be extended to \(p\)-adic formulas for products of Gauss sums in terms of multinomial coefficients. The paper contains several theorems relating \(p\)-adic Gauss sums to values of multinomial coefficients. The main ingredient of the proofs is the Gross-Koblitz formula expressing such Gauss sums in terms of products of values of Morita \(a\)-adic gamma functions. Combining these results with work of Lee and Hahn relating the \(p\)-adic Gauss sums to the \(p\)-adic value for the above \(a\), the author obtains values for \(a\) modulo \(p^s\) for any \(s>0\). Here is an example of such a result: Theorem 5.1. Suppose that \(p=7n+r\) is a prime with \(r\in \{1,2,4\}\) and write \(4p= a^2+7b^2\) with integers \(a,b\). Then for any positive integers \(m,s\) with \(m\equiv 1,2\), or \(4\pmod 7\), we have \[ \frac{a_{mp^s}} {a_{mp^{s-1}}} \equiv \frac{a+b\sqrt{-7}} {2}\pmod {p^s\mathbb Z_p}, \] where \((a_N)_{N\geq 1}\) is the sequence given by \(a_N={3k\choose k}\) if \(N=7k+1\), \(a_N=-{3k\choose k}\) if \(N=7k+2\), \(a_N={3k+1\choose k}\) if \(N=7k+4\) and \(a_N=0\) otherwise.
scientific article; zbMATH DE number 2081168