[Paper Review] Higher Rédei reciprocity and integral points on conics

Peter Koymans, Carlo Pagano|arXiv (Cornell University)|May 28, 2020

Algebraic Geometry and Number Theory28 references4 citations

TL;DR

This paper establishes an asymptotic formula for the density of squarefree integers $d$ for which the binary quadratic form $N_d(x,y) = l$ has integer solutions, where $l$ is a prime congruent to 3 modulo 4. Building on Smith's work and generalizing Rédei's reciprocity law, the authors derive a precise density $\gamma = \sum_{n=0}^\infty \frac{2^{-n^2} \eta_\infty \eta_n^{-2}}{2^{n+1} - 1}$, which captures the probability that the ideal above $l$ is the unique relation in the 2-torsion of the narrow class group of $\mathbb{Q}(\sqrt{d})$. This resolves a key case of Stevenhagen's conjecture on the negative Pell equation and extends results on the 2-primary part of class groups.

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 $\mathbb{Q}(\sqrt{d})$. Building on a breakthrough of Alexander Smith, 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 $\mathbb{Q}(\sqrt{-l}, \sqrt{d})$ is $2$ as $d$ varies over all positive squarefree integers. We also improve the results of Fouvry and Klüners and recent results of Chan, Milovic and the authors on the solubility of the negative Pell equation. Our main new tool is a generalization of a classical reciprocity law due to Rédei.

Motivation & Objective

• To determine the natural density of squarefree integers $d$ for which the equation $N_d(x,y) = l$ is soluble in integers, with $l$ a prime $\equiv 3 \pmod{4}$.
• To generalize Rédei's reciprocity law to study the distribution of the 2-primary part of narrow class groups of real quadratic fields.
• To establish a precise asymptotic formula for the proportion of such $d$ where the Hasse unit index of $\mathbb{Q}(\sqrt{-l}, \sqrt{d})$ is 2.
• To refine and improve prior results on the negative Pell equation and the 4-rank distribution of class groups.

Proposed method

• Generalize Rédei's reciprocity law to higher $2^n$-torsion subgroups of narrow class groups.
• Use Markov chain models to describe the distribution of the $4$-rank of $\operatorname{Cl}(\mathbb{Q}(\sqrt{d}))[2^\infty]$ as $d$ varies over squarefree integers divisible by $l$.
• Define the probability $\text{Pr}_{l,2}(n)$ that the $4$-rank of $\operatorname{Cl}(\mathbb{Q}(\sqrt{d}))[2^\infty]$ is $n$, and show it equals $\frac{1}{2^{n+1} - 1}$.
• Apply the Hasse–Minkowski theorem to reduce solubility over $\mathbb{Q}$ to local conditions, and use genus theory to analyze splitting behavior of $l$ in the genus field.
• Use the identity $\sum_{n=0}^\infty 2^{-n^2} \eta_\infty \eta_n^{-2} = 1$ to normalize the density measure and derive the final asymptotic.
• Prove the existence of the limit $\lim_{X \to \infty} \frac{|S_{\mathbb{Z},X,l}|}{|S_{\mathbb{Q},X,l}|} = \gamma$ via bounds on $\liminf$ and $\limsup$ using Markov chain dynamics.

Experimental results

Research questions

• RQ1 What is the natural density of squarefree integers $d$ for which $N_d(x,y) = l$ has integer solutions, with $l$ a prime $\equiv 3 \pmod{4}$?
• RQ2 How does the $4$-rank of the narrow class group $\operatorname{Cl}(\mathbb{Q}(\sqrt{d}))[2^\infty]$ influence the solubility of $N_d(x,y) = l$?
• RQ3 What is the asymptotic density of $d$ such that the Hasse unit index of $\mathbb{Q}(\sqrt{-l}, \sqrt{d})$ is 2?
• RQ4 To what extent does the generalized Rédei reciprocity law control the distribution of $2$-torsion in narrow class groups?
• RQ5 How does the probability that the prime $l$ corresponds to the unique relation in the $2$-torsion of the class group depend on the $4$-rank?

Key findings

• The asymptotic density of squarefree $d$ for which $N_d(x,y) = l$ is soluble in integers is $\gamma = \sum_{n=0}^\infty \frac{2^{-n^2} \eta_\infty \eta_n^{-2}}{2^{n+1} - 1}$, where $\eta_n = \prod_{j=1}^n (1 - 2^{-j})$.
• The probability that the $4$-rank of $\operatorname{Cl}(\mathbb{Q}(\sqrt{d}))[2^\infty]$ is $n$ is exactly $\frac{1}{2^{n+1} - 1}$, under the condition that $l \mid d$.
• The limit $\lim_{X \to \infty} \frac{|S_{\mathbb{Z},X,l}|}{|S_{\mathbb{Q},X,l}|} = \gamma$ exists and equals the stated infinite series, resolving a key case of Stevenhagen's conjecture on the negative Pell equation.
• The distribution of the $2$-primary part of the narrow class group of $\mathbb{Q}(\sqrt{d})$ as $d$ varies over squarefree integers divisible by $l$ is governed by a Markov chain model with transition probabilities derived from random linear maps and pairings over $\mathbb{F}_2$.
• The method provides a new proof of the asymptotic density for the event that the Hasse unit index of $\mathbb{Q}(\sqrt{-l}, \sqrt{d})$ is 2, when $l > 0$ is a prime $\equiv 3 \pmod{4}$, and confirms the prediction of Stevenhagen's conjecture in this case.
• The authors improve upon earlier results of Fouvry–Klüners and Chan–Milovic–Koymans–Pagano on the negative Pell equation by establishing a precise asymptotic density for the solubility of $x^2 - dy^2 = -1$ as $d$ varies over positive squarefree integers.