Extensions of classical congruences for parameters in binary quadratic forms (Q2773297)

Let \(\mathbb{Q}(\sqrt {-k})\) \((k\neq 3,4,8)\) be an imaginary quadratic field with discriminant \(-k\) and class number \(h\). Define NEWLINE\[NEWLINEU=\left \{u\in\mathbb{Z} \mid 0<u<k, \left({-k\over u}\right) =1\right\}NEWLINE\]NEWLINE and \(R={1\over k}\sum_{u\in U}u\). It is known that \(R={\varphi(k) \over 4}-{h\over 2}\in\mathbb{Z}.\)NEWLINENEWLINENEWLINEChoose integers \(t\) and \(w\) such that \(k=tw\), \(t\) (prime)\(>2\). Let \(p\) be a prime with \(({-k\over p})=1\). There are integers \(C\) and \(D\), unique up to sign, such that NEWLINE\[NEWLINE4p^h=C^2+kD^2,\;p\nmid C.NEWLINE\]NEWLINE The sign of \(C\) is determined uniquely by means of a congruence modulo \(t\) in such a way that \(C\) satisfies Stickelberger's congruence NEWLINE\[NEWLINEC\equiv\prod_{u\in U}[pu/k]!^{-1} \pmod p,NEWLINE\]NEWLINE where \([x]\) denotes the greatest integer \(\leq x\). The main goal of this paper is to determine \(C \pmod{p^2}\). This is accomplished by means of a non-routine application of the Gross-Koblitz formula for both \(p=2\) and \(p>2\). The excluded cases \(k=3\), 4, 8 are discussed separately.