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

Abstract

Let $p\\equiv 1\\hspace{0.2em}{\\rm mod}\\hspace{0.2em}4$\n 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)\xa0 $16$\n -rank of the class group $\\text{Cl}(-4p)$\n of the imaginary quadratic number field $\\mathbb{Q}(\\sqrt{-4p})$\n ; (ii)\xa0 $8$\n -rank of the ordinary class group $\\text{Cl}(8p)$\n of the real quadratic field $\\mathbb{Q}(\\sqrt{8p})$\n ; (iii)\xa0the solvability of the negative Pell equation $x^{2}-2py^{2}=-1$\n over the integers; (iv)\xa0 $2$\n -part of the Tate–Safarevic group $\\unicode[STIX]{x0428}(E_{p})$\n of the congruent number elliptic curve $E_{p}:y^{2}=x^{3}-p^{2}x$\n . Our results are conditional on a standard conjecture about short character sums.