Spins of prime ideals and the negative Pell equation $x^{2}-2py^{2}=-1

Let $p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) $16$ -rank of the class group $\text{Cl}(-4p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$ ; (ii) $8$ -rank of the ordinary class group $\text{Cl}(8p)$ of the real quadratic field $\mathbb{Q}(\sqrt{8p})$ ; (iii) the solvability of the negative Pell equation $x^{2}-2py^{2}=-1$ over the integers; (iv) $2$ -part of the Tate–Šafarevič group $\unicode[STIX]{x0428}(E_{p})$ of the congruent number elliptic curve $E_{p}:y^{2}=x^{3}-p^{2}x$ . Our results are conditional on a standard conjecture about short character sums.