Spins of prime ideals and the negative Pell equation

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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.