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

Abstract

Let $p\equiv 1\mathrm{mod}4$ 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 $\mathrm{Cl}(-4p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group $\mathrm{Cl}(8p)$ of the real quadratic field $\mathbb{Q}(\sqrt{8p})$; (iii) the solvability of the negative Pell equation $x^2-2p{y}^2=-1$ over the integers; (iv) $2$-part of the Tate-Šafarevič group 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.

27 pages, 1 field diagram; significant rewriting, including two novel applications and a new introduction

References in corpus (1)

• Counting primitive points of bounded height

Cited by in corpus (1)

• The $16$-rank of $\mathbb{Q}(\sqrt{-p})$