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

Part of: Additive number theory; partitions Multiplicative number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  23 November 2018

P. Koymans  and
D. Z. Milovic
P. Koymans
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands email p.h.koymans@math.leidenuniv.nl
D. Z. Milovic
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK email djordjo.milovic@ucl.ac.uk
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Abstract

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.

Keywords

class groupsnegative Pell equationsieve theory

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants 11R45: Density theorems 11N45: Asymptotic results on counting functions for algebraic and topological structures 11P21: Lattice points in specified regions
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 1 , January 2019 , pp. 100 - 125
Copyright
© The Authors 2018 

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Footnotes

The second author is supported by ERC grant agreement No. 670239.

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