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Redei reciprocity, governing fields and negative Pell

Published online by Cambridge University Press:  19 April 2021

PETER STEVENHAGEN
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands e-mail: psh@math.leidenuniv.nl
Corresponding
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Abstract

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation \[{x^2} - d{y^2} = - 1\].

Type
Research Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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