Higher Rédei reciprocity and integral points on conics

Abstract

Fix an integer l such that |l| is a prime 3 modulo 4. Let d > 0 be a squarefree integer and let N d (x, y) be the principal binary quadratic form of Q( √ d). Building on a breakthrough of Smith [33], we give an asymptotic formula for the solubility of N d (x, y) = l in integers x and y as d varies among squarefree integers divisible by l.As a corollary we give, in case l > 0, an asymptotic formula for the event that the Hasse Unit Index of the field Q( √ −l, √ d) is 2 as d varies over all positive squarefree integers. We also improve the results of Fouvry and Klüners [11,12] and recent results of Chan, Milovic and the authors [2] on the solubility of the negative Pell equation. Our main new tool is a generalization of a classical reciprocity law due to Rédei [31].