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arXiv:2201.13424math.NT

On Stevenhagen's conjecture

Peter Koymans, Carlo Pagano

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Abstract

We generalize a classical reciprocity law due to Rédei using our recently developed description of the $2$-torsion of class groups of multiquadratic fields. This result is then used to prove a variety of new reflection principles for class groups, one of which involves a symbol similar to the spin symbol as defined in work of Friedlander, Iwaniec, Mazur and Rubin. We combine these reflection principles with Smith's techniques to prove Stevenhagen's conjecture on the solubility of the negative Pell equation.

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math.NTNumber Theory

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  1. Mathematicians crack a simple but stubborn class of equations 1 comment

    • Right, this is about the negative Pell's equation, x^2 - dy^2 = -1 (rather than +1). The question is for which d this has a solution. (For the usual +1 case, which is what you're talking about, there are solutions for every d, and indeed continued fractions can be used.) There…

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