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

TL;DR

This paper investigates the distribution of certain algebraic invariants related to prime numbers congruent to 1 mod 4, using advanced number theoretic methods to analyze class groups, Pell equations, and elliptic curves, under a conjecture on character sums.

Contribution

It introduces a number field variant of Vinogradov's method to establish density results for multiple arithmetic invariants associated with primes congruent to 1 mod 4.

Findings

Density results for the 16-rank of class groups of imaginary quadratic fields.

Density results for the 8-rank of class groups of real quadratic fields.

Conditional solvability results for the negative Pell equation.

Abstract

Let $p\equiv 1\bmod 4$ 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 $x^2 - 2py^2 = -1$ over the integers; (iv) $2$-part of the Tate-\v{S}afarevi\v{c} group of the congruent number elliptic curve $E_p: y^2 = x^3-p^2x$. Our results are conditional on a standard conjecture about short character sums.