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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
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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 $x^2-dy^2=-1$ .

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

1. Introduction

In a 1939 Crelle paper [Reference Rédei16], the Hungarian mathematician László Rédei introduced a trilinear quadratic symbol $[a,b,c]\in\{\pm1\}$ for quadratic discriminants $a, b\in{\mathbf{Z}}$ and positive squarefree integers c satisfying a number of conditions. He used his symbol to describe 8-ranks of quadratic class groups, much in the way he had described the 4-ranks of these class groups in terms of Legendre symbols in his earlier work [Reference Rédei15]. His definition of the symbol, as a Jacobi symbol in the quadratic field ${\mathbf{Q}}(\sqrt a)$ , is somewhat involved, and seems to depend on many choices. Moreover, it only allows for a limited ‘symmetry’ of the symbol in its arguments, as infinite primes are disregarded in his definition.

An improved definition in class field theoretic terms was proposed in 2007 by Jens Corsman [Reference Corsman5]. He imposes fewer conditions on the arguments of the symbol, which are most conveniently taken in the group ${\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ of non-zero rational numbers modulo squares, requiring them to have relative quadratic Hilbert symbols

(1) \begin{equation}(a,b)_p=(a,c)_p=(b,c)_p=1\label{eqn1}\end{equation}

at all primes p, and to satisfy the coprimality condition

(2) \begin{equation}\gcd(\Delta(a),\Delta(b),\Delta(c))=1\label{eqn2}\end{equation}

for the discriminants of the associated quadratic fields, with $\Delta(1)=1$ . The Rédei symbol is then defined as a product $[a,b,c]=\prod_{p|c} [a,b,c]_p$ of ‘local’ symbols at the primes p dividing the squarefree integer representing c. It is essential to include the infinite prime $p=\infty$ in the product, with $\infty|c$ having the meaning $c<0$ . A correct definition of $[a,b,c]_p$ leads to a striking feature that we baptise Rédei’s reciprocity law.

Theorem 1·1. For $a, b, c \in{\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ satisfying (1) and (2), the Rédei symbol [a,b,c] is linear in each of its arguments, and satisfies the reciprocity law

\begin{equation*}[a,b,c]=[b,a,c]=[a,c,b].\end{equation*}

The Rédei symbol traditionally has its values in $\{\pm 1\}$ , as it may be computed as a Jacobi symbol in a quadratic field. Moreover, it has a definition as a product of local symbols that in (48) turn out to be quadratic Hilbert symbols, which invariably have values in $\{\pm 1\}$ , and satisfy a product formula on which the proof of Theorem 1·1 is based. However, the Rédei symbol is typically used in a linear algebra setting over the field of two elements ${\mathbf{F}}_2$ . It therefore makes sense to take its value in ${\mathbf{F}}_2$ , as we will do in our formal Definition 7·8.

Part of the perfect symmetry of the symbol [a,b,c] in its arguments is immediate from the definition, as [a,b,c] is the Artin symbol depending on c in a cyclic quartic extension $K={\mathbf{Q}}(\sqrt{ab})\subset F_{a,b}$ that depends symmetrically on a and b. Symmetry involving c is a true reciprocity: we may swap b and c in the symbol by a (non-obvious) application of quadratic reciprocity over ${\mathbf{Q}}(\sqrt a)$ .

The auxiliary field $F_{a,b}$ occurring in the definition of [a,b,c], which is only unique up to twisting by a finite group $T_{a,b}$ of quadratic characters (43), is the most complicated ingredient in the definition. Corsman failed to notice that $K\subset F_{a,b}$ may be ramified over 2, and that one has to require minimal ramification at 2 for $K\subset F_{a,b}$ in order for [a,b,c] to be well-defined, independently of the choice of $F_{a,b}$ .

In the case of prime arguments $a,b,c \equiv 1\bmod 4$ , no dyadic ramification subtleties arise, and the symbol has been interpreted by Morishita [13, section 8 $\cdot$ 2] as an arithmetic Milnor invariant, leading to a description as a triple Massey product that is useful in the study of pro-2-extensions of ${\mathbf{Q}}$ with given ramification locus [Reference Gärtner8].

Although Galois cohomology does play a role in Corsman’s approach to the Rédei symbol, its relation to Massey products and the applications of Rédei reciprocity to the average behavior of the 2-part of imaginary quadratic class groups in the recent work of Smith [Reference Smith18, Reference Smith19], neither the definition of the symbol nor the proof of its symmetry properties needs it, and we do not use it in this paper. Galois cohomology may be needed to find generalisations of the Rédei symbol. In fact, the linearity properties of the symbol make it one of the rare trilinear maps that ‘naturally occur’ in mathematics, and it is an interesting question whether the symbol has variants having properties of cryptographic interest in the sense of [Reference Boneh and Silverberg1].

In this paper we approach Rédei’s symbol along historical lines, showing how it arises in the study of the 2-part of the narrow class group C of a quadratic field K of discriminant D. Starting from old results in Section 2 on the 2-rank of C, we describe the 4-rank of C in terms of the Rédei matrix $R_4=R_4(D)$ , a matrix over the field ${\mathbf{F}}_2$ of 2 elements with entries that are essentially ${\mathbf{F}}_2$ -valued relative Legendre symbols of the primes dividing D (Theorem 3·1). Linear algebra also gives the 8-rank of C in terms of a matrix $R_8=R_8(D)$ over ${\mathbf{F}}_2$ (Theorem 4·1), but this time its entries are ( ${\mathbf{F}}_2$ -valued) Rédei symbols $[d_1, d_2, m]$ , given in Definition 4·4 as the Artin symbol of an ambiguous ideal in K of norm m in an unramified cyclic quartic extension $K={\mathbf{Q}}(\sqrt{d_1d_2})\subset F_{d_1, d_2}$ having ${\mathbf{Q}}(\sqrt {d_1}, \sqrt{d_2})$ as its intermediate quadratic extension. In Section 5, we explicitly compute $[d_1,d_2,m]$ in a field $F_{d_1, d_2}$ obtained by twisting the field F(x,y,z) in (33) associated to a primitive integral point on the conic

\begin{equation*}x^2-d_1y^2-d_2z^2=0\end{equation*}

by a quadratic character to ensure that $K\subset F_{d_1, d_2}$ is unramified at 2.

Section 6 shows how Rédei’s reciprocity law is suggested by the behaviour of small examples, and indicates how the general symbol [a, b,c] should be defined in order to obtain reciprocity. The precise definitions are in Section 7, leading to a proof of Theorem 1·1 in Section 8.

As an immediate application of Rédei reciprocity, Section 9 shows, following Corsman, how it yields the existence of governing fields for the 8-rank of class groups in 1-parameter families ${\mathbf{Q}}(\sqrt{dp})$ , with d a fixed integer and p a variable prime, a result that was originally obtained by different means in 1988 in [Reference Stevenhagen20], and that is at the basis of Smith’s work. Section 10 discusses its impact on my now classical Conjecture 3·2 on the number of real quadratic fields with fundamental unit of norm $-1$ or, equivalently, the asymptotic number of squarefree $d\in{\mathbf{Z}}_{>1}$ for which the negative Pell equation

(3) \begin{equation}x^2-dy^2=-1\label{eqn3}\end{equation}

is solvable in integers $x, y\in{\mathbf{Z}}$ .

2. The 2-rank

Let $d\ne 1$ be a squarefree integer, $K={\mathbf{Q}}(\sqrt d)$ the corresponding quadratic field, $D\in\{d, 4d\}$ the discriminant of K, and $C=Cl^+_K=Cl^+({\mathcal{O}}_K)$ the narrow class group of K, i.e., the quotient $C=I/P^+$ of the group I of fractional ${\mathcal{O}}_K$ -ideals by the subgroup of principal ideals $(x)=x{\mathcal{O}}_K$ with generator of positive norm N(x). The narrow class group maps surjectively to the ordinary class group $Cl_K$ of K, and we have an exact sequence

(4) \begin{equation}0\longrightarrow \langle F_\infty\rangle \longrightarrow C\longrightarrow Cl_K\to 0\label{eqn4}\end{equation}

in which $F_\infty$ , the Frobenius at $\infty$ , denotes the ideal class $[(\sqrt d)]\in C$ . This is the trivial element in C if K is imaginary quadratic, and also if K is real quadratic with fundamental unit $\varepsilon_d$ of norm $N(\varepsilon_d)=-1$ . If K is real quadratic with $N(\varepsilon_d)=1$ , then $F_\infty$ is of order 2, and C has twice the size of $Cl_K$ .

Describing the 2-part $C[2^\infty]\subset C$ consisting of all 2-power torsion elements in C can be done by specifying, for $k\ge1$ , the $2^k$ -rank

\begin{equation*}r_{2^k}=r_{2^k}(D)= \dim_{{{{\mathbf{F}}}_2}} C[2^k]/C[2^{k-1}]=\dim_{{{{\mathbf{F}}}_2}} 2^{k-1}C/2^kC.\end{equation*}

The sequence of non-negative integers $r_2, r_4, r_8, \ldots$ is non-increasing, and we have $r_{2^k}=0$ for k sufficiently large.

The 2-rank $r_2=\dim_{{{{\mathbf{F}}}_2}} C[2]=\dim_{{{{\mathbf{F}}}_2}} C/2C$ was already determined by Gauss, who defined C in terms of binary quadratic forms. To state his result, we factor D as a product

(5) \begin{equation}D=\prod_{i=1}^t p^*_i = t_D \prod_{p|D \text{ odd}} p^*\label{eqn5}\end{equation}

of signed odd prime discriminants $p^*=(-1)^{(p-1)/2}p\equiv 1\bmod 4$ and a discriminantal 2-part $t_D\in\{1, -4, \pm 8\}$ that we sloppily denote by $2^*$ in case D is even. We let ${\mathfrak{p}}_i|p_i$ be the prime of K lying over $p_i$ . It satisfies ${\mathfrak{p}}_i^2=(p_i)$ , so we have $[{\mathfrak{p}}_i]\in C[2]$ .

Theorem 2·1. We have $r_2=t-1$ , with t the number of prime divisors of D.

Proof. There are two fundamentally different proofs of this result, describing C[Reference Bosma and Stevenhagen2] and $C/2C$ , respectively. The first uses the t ambiguous ideal classes $[{\mathfrak{p}}_i]\in C[2]$ coming from the ramifying primes ${\mathfrak{p}}_i|p_i$ of K, the second the t genus characters $\chi_{p_i^*}\in\widehat C[2]$ corresponding to the discriminantal divisors $p^*_i$ in (5).

In the first proof, one exploits the Galois action on C of $Gal(K/{\mathbf{Q}})=\langle\sigma\rangle$ , noting that $\sigma$ acts by inversion as the norm map $N=1+\sigma$ annihilates C. A little Galois cohomology shows that the 2-torsion subgroup $C[2]=C[\sigma-1]$ is generated by the t classes $[{\mathfrak{p}}_i]$ , subject to a single relation. This yields $r_2=t-1$ .

For the second proof, one views $C=Gal(H/K)$ under the Artin isomorphism as the Galois group over K of the narrow Hilbert class field H of K. Then H is Galois over ${\mathbf{Q}}$ with dihedral Galois group

\begin{equation*}Gal(H/{\mathbf{Q}})\cong Gal(H/K)\rtimes Gal(K/{\mathbf{Q}})=C\rtimes \langle\sigma\rangle,\end{equation*}

as the surjection $Gal(H/{\mathbf{Q}})\to Gal(K/{\mathbf{Q}})=\langle\sigma\rangle$ is split and $\sigma$ acts by inversion. The genus field $H_2\subset H$ of K, which is defined as the maximal subfield of H that is abelian over ${\mathbf{Q}}$ , has as its Galois group over ${\mathbf{Q}}$ the elementary abelian 2-group

(6) \begin{equation}Gal(H_2/{\mathbf{Q}})=Gal(H/{\mathbf{Q}})^{ab} = C/2C \times \langle\sigma\rangle.\label{eqn6}\end{equation}

One can generate $H_2$ explicitly over ${\mathbf{Q}}$ by t independentquare roots as

(7) \begin{equation}H_2={\mathbf{Q}}(\{\sqrt {p_i^*} : i= 1,2, \ldots, t\}),\label{eqn7}\end{equation}

so $Gal(H_2/{\mathbf{Q}})$ is naturally an ${{{\mathbf{F}}}_2}$ -vector space of dimension t, in which the subspace $C/2C=Gal(H_2/K)\subset Gal(H_2/{\mathbf{Q}})$ has dimension $r_2=t-1$ .

The second proof of Theorem 2·1 shows that the prime power discriminants $p_i^*|D$ in (5) yield an ${\mathbf{F}}_2$ -basis of the quadratic characters on $Gal(H_2/{\mathbf{Q}})$ , with

(8) \begin{equation}\chi_{p_i^*}: Gal(H_2/{\mathbf{Q}}) \to Gal({\mathbf{Q}}(\sqrt {p_i^*})/{\mathbf{Q}})\cong {{{\mathbf{F}}}_2}\label{eqn8}\end{equation}

giving the Galois action on $\sqrt {p_i^*}$ . Even though this Galois action is given by multiplication by $\pm1$ , it is convenient for our linear algebra purposes to have additive characters with values in ${\mathbf{Q}}/{\mathbf{Z}}$ , and define the quadratic characters $\chi_{p_i^*}$ with values in $\frac{1}{2}{\mathbf{Z}}/{\mathbf{Z}}={\mathbf{F}}_2$ .

The character $\chi_{d_1}=\sum_{i\in S} \chi_{p_i}$ for a subset $S\subset \{1,2,\ldots, t\}$ corresponding to the discriminantal divisor $d_1=\prod_{i\in S} p_i^*$ of D gives the action on $\sqrt {d_1}$ . When restricted to $C/2C=Gal(H_2/ K)\subset Gal(H_2/{\mathbf{Q}})$ , it yields a quadratic character in the character group

(9) \begin{equation}\widehat C = Hom (C,{\mathbf{Q}}/{\mathbf{Z}})\label{eqn9}\end{equation}

of C that coincides with the character $\chi_{d_2}$ corresponding to the complementary divisor $d_2=D/d_1=\prod_{i\notin S} p_i^*$ . Rédei calls an unordered pair $(d_1, d_2)$ of quadratic discriminants satisfying

(10) \begin{equation}D=d_1d_2\label{eqn10}\end{equation}

a discriminantal decomposition of D. It ‘is’ the genus character $\chi_{d_1}=\chi_{d_2}\in \widehat C[2]$ , and the corresponding finitely unramified quadratic extension $K\subset E$ inside H is

(11) \begin{equation}E=K(\sqrt {d_1})={\mathbf{Q}}(\sqrt {d_1}, \sqrt {d_2})=K(\sqrt {d_2}).\label{eqn11}\end{equation}

3. The 4-rank and negative Pell

The ambiguous ideal proof of Theorem 2·1 describes the subgroup $C[2]\subset C$ as a quotient of ${\mathbf{F}}_2^t$ by a surjection

(12) \begin{equation}\alpha: {\mathbf{F}}_2^t \longrightarrow C[2]\label{eqn12}\end{equation}

that sends the jth basis vector to the class $[{\mathfrak{p}}_j]$ . The generator $A_D\in {\mathbf{F}}_2^t$ of its 1-dimensional kernel encodes the unique non-trivial relation that exists between the t classes of the ramifying primes ${\mathfrak{p}}_j|D$ of K. In particular, it tells us whether the element $F_\infty=[(\sqrt d)]\in C[2]$ in (4) is trivial. In the interesting case $D>0$ , this amounts to the fundamental unit of $K={\mathbf{Q}}(\sqrt D)$ having norm $N(\varepsilon_d)=-1$ .

The genus theory proof of Theorem 2·1 describes the quotient $C/2C=Gal(H_2/K)$ of C as a subspace of $Gal(H/{\mathbf{Q}})={\mathbf{F}}_2^t$ under the inclusion map

(13) \begin{equation}\gamma: C/2C=Gal(H_2/K)\longrightarrow Gal(H_2/{\mathbf{Q}})={\mathbf{F}}_2^t,\label{eqn13}\end{equation}

with the ith coordinate of $\gamma({\mathfrak{a}})\in {\mathbf{F}}_2^t$ for $[{\mathfrak{a}}]\in C$ describing the action of the Artin symbol $Art({\mathfrak{a}}, H/K) \in Gal(H/K)$ on $\sqrt{p_i^*}$ . As elements of $Gal(H_2/K)$ fix the product $\prod_{i=1}^t\sqrt{p_i^*}=\sqrt D$ , the map $\gamma$ embeds $C/2C$ as the ‘sum-zero-hyperplane’ in ${\mathbf{F}}_2^t$ . Equivalently, one can formulate this as in (12) by saying that the subgroup $\widehat C[2]\subset \widehat C$ of quadratic characters on C is generated by the t characters $\chi_{p_i^*}$ , subject to the relation that their sum

(14) \begin{equation}\chi_D= \sum_{i=1}^t \chi_{p_i^*},\label{eqn14}\end{equation}

the Dirichlet character corresponding to K, is the character on $Gal(H_2/{\mathbf{Q}})$ in (6) that has kernel $C/2C$ and is trivial as an element of $\widehat C$ . This time, the relation holds no deeper information as it is ‘the same’ for all quadratic fields.

The 4-rank of C is the ${{{\mathbf{F}}}_2}$ -dimension of the kernel $C[2]\cap 2C$ of the natural map

\begin{equation*}\varphi_4: C[2]\longrightarrow C/2C,\end{equation*}

and we can find it by combining $\varphi_4$ with the surjection $\alpha$ and the injection $\gamma$ from (12) and (13) into a single ${{{\mathbf{F}}}_2}$ -linear Rédei map

(15) \begin{equation}R_4:\quad{\mathbf{F}}_2^t \mathop{\longrightarrow}\limits^{{\alpha}} C[2] \mathop{\longrightarrow}\limits^{\varphi_4} C/2C \mathop{\longrightarrow}\limits^{{\gamma}} Gal(H_2/{\mathbf{Q}})={\mathbf{F}}_2^t.\label{eqn15}\end{equation}

We have

\begin{equation*}1+r_4=1+\dim_{{{{\mathbf{F}}}_2}} \ker \varphi_4=\dim_{{{{\mathbf{F}}}_2}} \ker R_4 = t-rank_{{{{\mathbf{F}}}_2}} R_4,\end{equation*}

and writing $r_2=t-1$ as in Theorem 3·1, we obtain the following result.

Theorem 3·1. The 4-rank of C equals $r_4 = r_2 - rank_{{{{\mathbf{F}}}_2}} R_4$ .

Explicit entries for the matrix $R_4=(\varepsilon_{ij})_{i, j}\in Mat_{t \times t}({{{\mathbf{F}}}_2})$ can easily be given. The entry $\varepsilon_{ij}$ describes the action of the Artin symbol $Art ({\mathfrak{p}}_j, H/K)$ on $\sqrt{p_i^*}\in H_2\subset H$ . For $i\ne j$ , it is an ${\mathbf{F}}_2$ -valued Legendre (or for $p_j=2$ Kronecker) symbol:

(16) \begin{equation}\varepsilon_{ij}=\chi_{p_i^*}([{\mathfrak{p}}_j])={\left(\frac{p_i^*}{p_j}\right)}\in{\mathbf{F}}_2.\label{eqn16}\end{equation}

The diagonal entries $\varepsilon_{jj}=\sum_{i\ne j} \varepsilon_{ij}$ of $R_4$ follow from the sum-zero-property of $\gamma$ : the rows of $R_4$ add up to $0\in{\mathbf{F}}_2^t$ . This simple description of $r_4$ in terms of the relative quadratic behavior of the primes $p_i$ dividing D goes back to Rédei [Reference Rédei15].

The somewhat hybrid notation in (16) uses an identification ‘ $\{\pm1\}={\mathbf{F}}_2$ ’ of multiplicative and additive value groups of quadratic symbols and characters. The same notational ambiguity inevitably occurs for Rédei symbols, which are quadratic symbols in quadratic fields, with values that are traditionally taken in $\{\pm1\}$ , but that also occur as entries of matrices over ${\mathbf{F}}_2$ . We have mostly chosen additive values of characters and symbols in this paper, but multiplicative values are used in the proof of Rédei’s reciprocity law in Section 8, which relates Rédei symbols to quadratic Hilbert symbols.

For $K={\mathbf{Q}}(\sqrt d)$ real quadratic of discriminant $D\in\{d, 4d\}$ , the fundamental unit has norm $N(\varepsilon_d)=-1$ if and only if the negative Pell equation

(17) \begin{equation}x^2-dy^2=-1\label{eqn17}\end{equation}

is solvable in integers $x, y\in{\mathbf{Z}}$ . If it exists, the smallest solution to (17) can be found from the continued fraction expansion of $\sqrt d$ , which then needs to have odd period length, or from general unit finding algorithms in number rings [9]. For $d=D\equiv 5\bmod 8$ , this solution corresponds to the cube of $\varepsilon_d$ in case the fundamental unit $\varepsilon_d\in{\mathcal{O}}_K={\mathbf{Z}}[(1+\sqrt d)/2]$ does not lie in ${\mathbf{Z}}[\sqrt d]$ . Conjecturally [Reference Stevenhagen22], this happens for a fraction $2/3$ of squarefree $D\equiv 5\bmod 8$ .

For solvability of (17) in rational numbers, which amounts to K having elements of norm $-1$ or, equivalently, having quadratic Hilbert symbols

(18) \begin{equation}(d,-1)_p=(D, -1)_p=1\qquad\hbox{for all primes $p\le\infty$,}\label{eqn18}\end{equation}

there is an easy criterion: solvability occurs if and only if d (or D) is positive and without prime factors $p\equiv 3\bmod 4$ . By [17, Satz 3], the set ${\mathcal{D}}$ of such D is a thin set, asymptotically containing $cX/\sqrt{\log X}$ elements $D<X$ , for some explicit $c\approx .348\in {\mathbf{R}}_{>0}$ . For $D\in {\mathcal{D}}$ we have $t_D\in\{1, 8\}$ and $p^*=p$ in (5), and by (7)

(19) \begin{equation}D \in {\mathcal{D}}\quad\Longleftrightarrow\quad H_2 \hbox{ is totally real.}\label{eqn19}\end{equation}

The class field theoretic implication of (4) is that the set ${{\mathcal{D}}}^-\subset {\mathcal{D}}$ of discriminants for which the negative Pell equation (17) is solvable in integers has a similar description:

(20) \begin{equation}D \in {\mathcal{D}}^-\quad\Longleftrightarrow\quad H \hbox{ is totally real.}\label{eqn20}\end{equation}

Indeed, the narrow Hilbert class field H of $K={\mathbf{Q}}(\sqrt d)$ is totally real if and only if the Frobenius at infinity $F_\infty\in C$ at the real primes of $K={\mathbf{Q}}(\sqrt d)$ is trivial on H.

Assume $D\in {\mathcal{D}}$ . Then the map $\alpha: {\mathbf{F}}_2^t\to C[2]$ in (12) describes $F_\infty=[(\sqrt d)]$ as

\begin{equation*}F_\infty=\alpha[(1)_{i=1}^t] \in C[2]\cap 2C,\end{equation*}

and $V=\alpha^{-1}(C[2]\cap 2C)\subset {\mathbf{F}}_2^t$ is an $(r_4+1)$ -dimensional subspace containing $(1)_{i=1}^t$ . The linear map $\alpha|_V: V\to C[2]\cap 2C$ is surjective with kernel ${\mathbf{F}}_2\cdot A_D$ , and numerical evidence [Reference Bosma and Stevenhagen2] suggests that $A_D$ behaves like a ‘random’ non-zero element in V. As $D\in{\mathcal{D}}$ satisfies

\begin{equation*}D\in {{\mathcal{D}}}^-\Longleftrightarrow A_D=(1)_{i=1}^t,\end{equation*}

we expect that for the discriminants $D\in{\mathcal{D}}$ having 4-rank $r_4=e$ , the negative Pell equation (17) will be solvable with ‘probability’ $(\#V-1)^{-1}=(2^{r_4+1}-1)^{-1}$ .

To heuristically find the density for ${\mathcal{D}}^-$ in ${\mathcal{D}}$ , write ${\mathcal{D}}=\bigcup_{e=0}^\infty {\mathcal{D}}(e)$ , with ${\mathcal{D}}(e)$ the subset of $D\in {\mathcal{D}}$ having 4-rank $r_4=e$ . By Theorem 3·1, a discriminant $D\in{\mathcal{D}}$ is in ${\mathcal{D}}(e)$ if and only if its Rédei matrix $R_4\in Mat_t({\mathbf{F}}_2)$ has corank $e+1$ . Quadratic reciprocity for the entries (16) implies that the matrix $R_4$ is symmetric, with rows and columns adding up to 0. The (1,1)-minor of $R_4$ , which determines the full matrix $R_4$ , behaves as a random symmetric $(t-1)\times(t-1)$ -matrix. As the average number t of primes factors of D tends to infinity with D, albeit very slowly, as $\log\log D$ , we expect the density of ${\mathcal{D}}(e)$ in ${\mathcal{D}}$ to equal $P(e)=\lim_{n\to\infty}P_n(e)$ , with $P_n(e)$ the fraction of the ${n+1\choose 2}$ symmetric $n\times n$ -matrices over ${\mathbf{F}}_2$ having corank e. In terms of the infinite product

(21) \begin{equation}\alpha = \prod_{j\ \text{odd}} (1-2^{-j})=\prod_{j=1}^\infty (1+2^{-j})^{-1}\approx .419422441,\label{eqn21}\end{equation}

we have $P(0)=\alpha$ , and more generally $P(e)=\alpha\cdot \prod_{j=1}^e (2^j-1)^{-1}$ for $e\ge0$ .

By the probability argument above, we expect the natural density of ${\mathcal{D}}^-(e)$ in ${\mathcal{D}}(e)$ to be $1/(2^{e+1}-1)$ for all $e\in{\mathbf{Z}}_{\ge0}$ . This leads to my 1992 conjecture for the solvability of the negative Pell equation [Reference Stevenhagen21].

Conjecture 3·2. The set ${\mathcal{D}}^-$ of discriminants of quadratic fields with fundamental unit of norm $-1$ has natural density

\begin{equation*}\sum_{e=0}^\infty \frac{P(e)}{2^{e+1}-1} =1-\alpha \approx .580577559\end{equation*}

inside the set ${\mathcal{D}}$ of discriminants of quadratic fields containing elements of norm $-1$ .

Fouvry and Klüners proved in 2010 [Reference Fouvry and Klüners6] that ${\mathcal{D}}(e)$ does indeed have the expected density P(e) in ${\mathcal{D}}$ . For $e=0$ we have $2^{e+1}-1=1$ and ${\mathcal{D}}(0)\subset {\mathcal{D}}^-$ , as in the case $r_4=0$ the narrow Hilbert class field H is totally real, being a normal number field of odd degree over the totally real genus field $H_2$ . This immediately implies that $P(0)=\alpha$ is a lower bound for the lower density of ${\mathcal{D}}^-$ in ${\mathcal{D}}$ . To get better bounds, one needs control over the archimedean character of H for $e\ge1$ . We will address this in Section 10.

4. The 8-rank

The 8-rank $r_8$ of C equals the ${{{\mathbf{F}}}_2}$ -dimension of the kernel of the natural map

\begin{equation*}\varphi_8: C[2]\cap 2C \longrightarrow 2C/4C\end{equation*}

between $r_4$ -dimensional vector spaces over ${{{\mathbf{F}}}_2}$ . Under the Artin isomorphism, the group $2C/4C$ is the Galois group $Gal(H_4/H_2)$ , with $H_4\subset H$ the narrow 4-Hilbert class field of K. We can restrict $\alpha$ in (12) to the kernel of the 4-rank map $R_4$ from (15) and compose with $\varphi_8$ to obtain an ${{{\mathbf{F}}}_2}$ -linear map

(22) \begin{equation}R_8:\quad\ker R_4 \mathop{\longrightarrow}\limits^{{\alpha}} C[2]\cap 2C \mathop{\longrightarrow}\limits^{{\varphi_8}} 2C/4C=Gal(H_4/H_2)\cong {\mathbf{F}}_2^{r_4}\label{eqn22}\end{equation}

defined on the $(r_4+1)$ -dimensional space $\ker R_4$ . Here we write $Gal(H_4/H_2)\cong {\mathbf{F}}_2^{r_4}$ , in contrast to the equality $Gal(H_2/{\mathbf{Q}})={\mathbf{F}}_2^t$ in (15), as we no longer have an obvious choice for the basis of this ${\mathbf{F}}_2$ -vector space. As $r_8 = \dim_{{{{\mathbf{F}}}_2}} \ker \varphi_8$ is the codimension of the image of $\varphi_8$ in $2C/4C$ , we obtain the following analogue of Theorem 3·1.

Theorem 4·1. The 8-rank of C equals $r_8 = r_4-rank_{{{{\mathbf{F}}}_2}} R_8$ .

In order to obtain a matrix representing $R_8$ , we want to represent $Gal(H_4/H_2)$ in (22) as explicitly as we represented $Gal(H_2/{\mathbf{Q}})$ in (15). Rédei had already achieved this in a 1934 paper with Reichardt [Reference Rédei and Reichardt14], which computed $r_4$ not as in Theorem 3·1 but by an explicit construction of $H_2\subset H_4$ in terms of $r_4$ cyclic quartic extensions $K\subset F$ inside H that are K-linearly disjoint. For such unramified F, the group $Gal(F/{\mathbf{Q}})$ is dihedral of order 8, and the intersection $E=F\cap H_2$ equals $E = {\mathbf{Q}}(\sqrt {d_1}, \sqrt {d_2})$ for a discriminantal decomposition $D=d_1d_2$ as in (10) and (11).

Rédei calls the decompositions $D=d_1d_2$ defining those $E={\mathbf{Q}}(\sqrt {d_1}, \sqrt {d_2})$ that arise as $F\cap H_2$ zweiter Art, ‘of the second kind’. For the corresponding quadratic character $\chi\in \widehat C =Hom (C,{\mathbf{Q}}/{\mathbf{Z}})$ , it means that we have $\chi=2\psi$ for a quartic character $\psi$ defining $K\subset F$ . By the duality of finite abelian groups, we have $\chi\in 2\widehat C$ if and only if $\chi$ vanishes on the 2-torsion subgroup C[Reference Bosma and Stevenhagen2]. This leads to the following characterization of these quadratic characters.

Lemma 4.·2. For a quadratic character $\chi\in \widehat C[2]$ defining $E={\mathbf{Q}}(\sqrt{d_1},\sqrt{d_2})$ as in (10) and (11), having $\chi\in 2\widehat C$ is equivalent to each of the following :

  1. (i) there exists a cyclic quartic extension $K\subset F$ inside H containing E ;

  2. (ii) $\chi_{d_1}\circ R_4=\chi_{d_2}\circ R_4$ is the zero map ;

  3. (iii) all ramified primes of K split completely in $K\subset E$ ;

  4. (iv) for $i=1,2$ and $p|d_i$ prime we have $\big(\tfrac{D/d_i}{p}\big)=1$ .

Proof. Having $\chi=2\psi$ for a quartic character $\psi\in\widehat C$ defining F as in (i) is equivalent to $\chi$ vanishing on the subgroup C[2] of ambiguous ideal classes generated by the classes of the ramifying primes ${\mathfrak{p}}|D=d_1d_2$ of K as in (12). One can phrase this using the map $R_4$ from (15) as in (ii), or in terms of the splitting of the ramifying primes in $K\subset E$ as in (iii). A ramifying prime of K divides exactly one of $d_1$ , $d_2$ . A prime ${\mathfrak{p}}|p$ in K dividing, say, $d_1$ splits completely in $K\subset E=K(\sqrt{d_2})$ if and only if the Legendre (or Kronecker) symbol $\big(\tfrac{d_2}{p}\big)$ equals 1, as in (iv).

Remark 4.·3. The identity $\chi=2\psi$ determines $\psi\in\widehat C$ up to a quadratic character, as an element of $\widehat C[4]/\widehat C[2]$ , and this means that the quadratic extension $E\subset F$ it gives rise to in (i) is determined by $\chi$ only up to ‘twisting’ by an unramified quadratic character. In other words: not $E\subset F$ , but the quadratic extension $H_2\subset H_2F$ it generates over the genus field $H_2$ is unique.

In order to compute the 8-rank in Theorem 4·1 from the rank of an explicit matrix describing the map $R_8$ in (22), we choose an ${{{\mathbf{F}}}_2}$ -basis for the $(r_4+1)$ -dimensional subspace $\ker R_4\subset {\mathbf{F}}_2^t$ , and write

\begin{equation*}[{\mathfrak{m}}_j]\in C[2]\cap 2C\qquad\quad (j=1,2,\ldots, r_4+1)\end{equation*}

for the images of these basis vectors under the map $\alpha$ from (12). The classes $[{\mathfrak{m}}_j]$ span $C[2]\cap 2C$ , subject to a single relation encoded in $A_D$ , and their Artin symbols $Art ({\mathfrak{m}}_j, H/K)$ are the identity on the genus field $H_2$ .

Similarly, we choose quartic characters $\psi_i$ for $i=1,2,\ldots, r_4$ spanning $\widehat C[4]/\widehat C[2]$ , and let $K\subset F_{4,i}$ be the corresponding unramified quartic extensions. By condition (ii) of Lemma 4.2, the quadratic characters $\chi_i=2\psi_i\in\widehat C[2]$ come from vectors in ${\mathbf{F}}_2^t$ that, together with $(1)_{i=1}^t$ , span the kernel of the transpose $R_4^T$ of the Rédei matrix in (15). The $r_4$ quadratic extensions $H_2\subset H_2F_{4,i}$ span $H_2\subset H_4$ , and the map $R_8$ is represented by a matrix $R_8=(\eta_{ij})_{i,j}\in Mat_{r_4\times (r_4+1)}({{{\mathbf{F}}}_2})$ with entries

(23) \begin{equation}\eta_{ij}= \psi_i[{\mathfrak{m}}_j]=Art ({\mathfrak{m}}_j, H_2F_{4,i}/K) \in Gal(H_2F_{4,i}/H_2)={{{\mathbf{F}}}_2}.\label{eqn23}\end{equation}

In cases where we know the kernel of $\alpha$ in (12), i.e., the non-trivial relation $A_D$ between the ramified primes of K in C, we can use it to leave out a column of $R_8$ corresponding to a ‘superfluous’ generator $[{\mathfrak{m}}_j]$ of $C[2]\cap 2C$ , and work with an $(r_4\times r_4)$ -matrix to describe $\varphi_8$ in (22).

A product ${\mathfrak{m}}$ of distinct ramified primes of K is characterized by the squarefree divisor $m|D$ arising as its norm, and the residue class of a quartic character $\psi$ in $\widehat C[4]/\widehat C[2]$ by the invariant field $E={\mathbf{Q}}(\sqrt{d_1}, \sqrt{d_2})$ of the quadratic character $2\psi$ corresponding to a decomposition $D=d_1d_2$ ‘of the second kind’. This leads to a classical notation for the entries $\psi([{\mathfrak{m}}])$ in (23) as Rédei symbols.

Definition 4.·4. Let $D=d_1d_2$ be a decomposition of the second kind, $K\subset F$ a corresponding extension as in condition (i) of Lemma 4.2, and $m|D$ the squarefree norm of an integral ideal ${\mathfrak{m}}$ in K with $[{\mathfrak{m}}]\in C[2]\cap 2C$ . Then the Rédei symbol associated to $d_1$ , $d_2$ , and m is the Artin symbol

\begin{equation*}[d_1, d_2, m]=Art ({\mathfrak{m}}, H_2F/K)\in Gal(H_2F/H_2)={{{\mathbf{F}}}_2}.\end{equation*}

It is convenient to take the value of Rédei symbols in ${{{\mathbf{F}}}_2}$ , as we do in Definition 4·4. After all, they arise as entries $\eta_{ij}$ of an ${{{\mathbf{F}}}_2}$ -matrix $R_8$ in (23). However, they describe the Galois action on certain square roots, just like the entries $\varepsilon_{ij}$ of $R_4$ in (16), so their values are traditionally taken in $\{\pm1\}$ . We have been unable to completely avoid this notational ambiguity, which already occurs in (16). Despite our additive definition, our proof of Rédei reciprocity in Section 8 views Rédei symbols in (46) as ‘products’ of local symbols $[a,b,c]_p$ , which are recognised in our key Lemma 8·1 as quadratic Hilbert symbols satisfying a well-known global product formula that we did not rename into a sum formula. On the other hand, our quadratic characters in (8) and biquadratic characters in (38) do take values in ${{{\mathbf{F}}}_2}$ .

5. Computing Redei-symbols

Definition 4·4 of the Rédei symbol $[d_1, d_2, m]$ does not immediately show how to compute it from $d_1$ , $d_2$ and m, as it involves a quadratic extension F of ${\mathbf{Q}}(\sqrt{d_1}, \sqrt{d_2})$ that is dihedral over ${\mathbf{Q}}$ . Galois theory tells us that such F come from rational points on the conic $x^2-d_1y^2-d_2z^2=0$ , and that they are unique up to twisting by quadratic characters. This statement does not depend on the base field ${\mathbf{Q}}$ , and we formulate it for any field Q of characteristic different from 2.

Lemma 5·1. Let Q be of characteristic different from 2, and $Q\subset Q(\sqrt a)$ a quadratic extension. For $\beta\in Q(\sqrt a)^*$ non-square of norm $N\beta=b\in Q^*$ , let F be the normal closure of the quartic extension $Q(\sqrt a,\sqrt \beta)$ of Q. Then :

  1. (i) for $\bar b\notin\{\bar 1, \bar a\}\subset Q^*/{Q^*}^2$ , the field F is quadratic over $Q(\sqrt a, \sqrt b)$ , cyclic of degree 4 over $Q(\sqrt {ab})$ , and dihedral of degree 8 over Q ;

  2. (ii) for $\bar b=\bar a\in Q^*/{Q^*}^2$ , the field F is quadratic over $Q(\sqrt a)$ and cyclic of degree 4 over Q ;

  3. (iii) for $\bar b=\bar 1\in Q^*/{Q^*}^2$ , the field F is quadratic over $Q(\sqrt a)$ and non-cyclic abelian of degree 4 over Q.

Conversely, every field F having the properties in (i), (ii) or (iii) is obtained in this way for some $\beta\in Q(\sqrt a)$ of norm b.

Proof. Basic Galois theory.

Corollary 5·2 Let $a, b\in Q^*$ and $E=Q(\sqrt a, \sqrt b)$ be as in (i) of Lemma 5·1. Then a quadratic extension $E\subset F$ is cyclic over $Q(\sqrt{ab})$ and dihedral of degree 8 over Q if and only if there exists a non-zero solution $(x,y,z)\in Q^3$ to the equation

\begin{equation*}x^2-a y^2-b z^2=0\end{equation*}

such that for $\beta=x+y\sqrt a\in Q(\sqrt a)$ and $\alpha=2(x+z\sqrt b)\in Q(\sqrt b)$ of norm $\beta\beta'\in b\cdot {Q^*}^2$ and $\alpha\alpha'\in a \cdot {Q^*}^2$ , we have

\begin{equation*}F=E(\sqrt \beta)=E(\sqrt\alpha).\end{equation*}

Given $F=E(\sqrt \beta)$ , any other quadratic extension of E that is dihedral over Q is of the form $F_t=E(\sqrt{t\beta})$ for some unique $t\in Q^*/\langle a, b, {Q^*}^2\rangle$ .

Proof. The first statement follows if we write $\beta=x+y\sqrt a\in Q(\sqrt a)$ in the dihedral case (i) of Lemma 5·1, and observe that $F=Q(\sqrt a, \sqrt \beta, \sqrt{\beta'})=E(\sqrt \beta)$ is the normal closure over Q of $Q(\sqrt a,\sqrt \beta)$ , but also of $Q(\sqrt b,\sqrt\alpha)$ : it contains a square root of the non-square element

(24) \begin{equation}(\sqrt\beta+\sqrt{\beta'})^2=\left(\sqrt{x+y\sqrt a}+\sqrt{x-y\sqrt a}\right)^2= 2(x+ z\sqrt{b})=\alpha \in {\mathbf{Q}}(\sqrt{b})^*.\label{eqn24}\end{equation}

The dihedral group $D_4$ of order 8 has center $Z(D_4)={\mathbf{F}}_2$ with quotient $D_4/Z(D_4)={\mathbf{F}}_2\times{\mathbf{F}}_2$ , and extending $Q\subset Q(\sqrt a, \sqrt b)$ to a $D_4$ -extension amounts to lifting the surjection $G_Q\to{\mathbf{F}}_2\times{\mathbf{F}}_2=D_4/Z(D_4)$ on the absolute Galois group of Q corresponding to $Q(\sqrt a, \sqrt b)$ to a homomorphism $f:G_Q\to D_4$ . Given f corresponding to $F=E(\sqrt \beta)$ , any other lift is of the form $f_t=\chi_t f$ for some quadratic character $\chi_t: G_Q\to Z(D_4)={\mathbf{F}}_2$ corresponding to $Q\subset Q(\sqrt t)$ . The extension $F_t$ corresponding to $f_t$ is the quadratic twist $F_t=E(\sqrt{t\beta})$ of F, and $t\in Q^*$ yielding $F_t$ is unique up to multiplication by elements of ${E^*}^2\cap Q^*=\langle a, b, {Q^*}^2\rangle$ .

Corollary 5·3 For $E=Q(\sqrt a)$ as in Lemma 5·1, a quadratic extension $E\subset F$ is cyclic over Q if and only if there exists a non-zero solution $(x,y,z)\in Q^3$ to

\begin{equation*}x^2-a y^2-a z^2=0\end{equation*}

such that we have $F=E(\sqrt\alpha)$ for $\alpha=x+y\sqrt a\in Q(\sqrt a)$ of norm $\alpha\alpha'\in a\cdot {Q^*}^2$ . Given one $F=E(\sqrt \alpha)$ , any other such extension is of the form $F_t=E(\sqrt{t\alpha})$ for some unique $t\in Q^*/\langle a, {Q^*}^2\rangle$ .

Proof. Analogous to the dihedral case.

Remark 5·4. The dihedral group $D_4$ of order 8 can be viewed as the Heisenberg group $U_3({{{\mathbf{F}}}_2})$ of upper triangular $3\times 3$ -matrices with coefficients in ${{{\mathbf{F}}}_2}$ , and extending an extension $Q\subset Q(\sqrt a, \sqrt b)$ to a $D_4$ -extension amounts to an embedding problem that can be treated in terms of Massey symbols [Reference Mináć and Duy Tân12]. For our purposes, the basic Galois theory of Lemma 5·1 and its corollaries are already sufficient.

In order to construct an unramified extension $K\subset F$ containing $E={\mathbf{Q}}(\sqrt{d_1}, \sqrt{d_2})$ for $D=d_1d_2$ satisfying the conditions of Lemma 4.2, we apply Corollary 5·2 for $Q={\mathbf{Q}}$ and $(a,b)=(d_1, d_2)$ . It shows that F can be explicitly generated as

(25) \begin{equation}F=F(x,y,z)=E(\sqrt {\delta_2})=E(\sqrt {\delta_1}),\label{eqn25}\end{equation}

for elements $\delta_2=x+y\sqrt{d_1}\in {\mathbf{Q}}(\sqrt{d_1})^*$ and $\delta_1=2(x+y\sqrt{d_1})\in {\mathbf{Q}}(\sqrt{d_1})^*$ coming from a solution (x,y,z) to the equation

(26) \begin{equation}x^2-d_1y^2-d_2 z^2=0.\label{eqn26}\end{equation}

By Corollary 5·2, scaling any non-zero solution (x,y,z) with an appropriate element $t\in{\mathbf{Q}}^*$ , which amounts to replacing F(x,y,z) by the quadratic twist F(tx,ty,tz), will make $K\subset F$ unramified. As we will show in a slightly more general setting in Corollary 3, every primitive integral solution (x,y,z) to (26) yields an extension $K\subset F(x,y,z)$ that is unramified at all odd primes. Ramification over 2 can be avoided by twisting the extension with a suitable choice of $t\in \{\pm1, \pm2\}$ .

Example 5·5. Take $K={\mathbf{Q}}(\sqrt{-205})$ of discriminant $D=-4\cdot5\cdot41=-820$ , which has $t=3$ and $r_2=2$ . The columns of the Rédei matrix

\begin{equation*}R_4=\begin{pmatrix}1 &\quad 0 &\quad 0\\ 1 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0\end{pmatrix}\end{equation*}

describe the action of the Artin symbols of the three ramified primes ${\mathfrak{p}}_2$ , ${\mathfrak{p}}_5$ , and ${\mathfrak{p}}_{41}$ dividing D on the square roots of $-4$ , 5 and 41 generating $H_2={\mathbf{Q}}(i, \sqrt 5, \sqrt {41})$ as in (16). From the matrix $R_4$ we read off that $r_4$ equals $r_2-rank(R_4)=1$ , that $[{\mathfrak{p}}_5]$ and $[{\mathfrak{p}}_{41}]$ span $C[2]\cap 2C$ , and that $D=-20\cdot41$ is the unique decomposition of the second kind. The equation

\begin{equation*}x^2+20y^2-41z^2=0\end{equation*}

has a primitive solution (12,1,2) for which the element $\delta=12+2\sqrt{-5}$ of norm $2^2\cdot 41$ is ‘primitive outside 2’ and satisfies $\delta\equiv(1+\sqrt{-5})^2\bmod 4$ . This shows that $\delta=2(6+\sqrt{-5})$ has an unramified square root over $E={\mathbf{Q}}(\sqrt{-5}, \sqrt{41})$ , whereas the primitive elements $\pm6+\sqrt{-5}$ yield extensions $E\subset E(\sqrt{\pm\delta/2})$ that are ramified over 2.

The solution (17,2,3) defining the primitive element $\delta_0=17+4\sqrt{-5}$ of norm $3^2\cdot 41$ satisfying $\delta_0\equiv1\bmod4$ also has an unramified square root over E. We have

\begin{equation*}\delta\delta_0 = 164+82\sqrt{-5} = -[\sqrt{41}(1-\sqrt{-5})]^2\in -1\cdot {E^*}^2,\end{equation*}

and $E(\sqrt{\delta_0})=E(\sqrt{t\delta})$ for $t=-1$ . Over $H_2$ , both $\sqrt{\delta_0}$ and $\sqrt{\delta}$ generate

\begin{equation*}H_4=H_2(\sqrt{\delta_0})=H_2(\sqrt{\delta}).\end{equation*}

As we know that $(\sqrt{-5\cdot 41})={\mathfrak{p}}_5{\mathfrak{p}}_{41}$ is trivial in C, the class of either ${\mathfrak{p}}_5$ or ${\mathfrak{p}}_{41}$ generates $C[2]\cap 2C$ . The matrix $R_8$ consists of a single Rédei symbol

\begin{equation*}[-20, 41, 5]=[-20,41, 41]\end{equation*}

describing whether the prime ${\mathfrak{p}}_5$ (or, equivalently, ${\mathfrak{p}}_{41}$ ) of K splits completely in $H_4$ . It does not, as $\delta=12+2\sqrt{-5}$ (like $\delta_0=17+4\sqrt{-5})$ ) is congruent to the quadratic non-residue 2 modulo every prime over 5 in $H_2$ . We conclude that we have $r_8=0$ , and that the 2-part of C is isomorphic to ${\mathbf{Z}}/2{\mathbf{Z}}\times {\mathbf{Z}}/4{\mathbf{Z}}$ .

In this case, the decomposition $D=d_1d_2=-4\cdot 205$ is not of the second kind, but the conic

\begin{equation*}x^2+4y^2-205z^2=0\end{equation*}

defined by (26) does have infinitely many rational points (x,y,z), such as (3,7,1). None of them defines an unramified quartic extension $K\subset F(x,y,z)$ .

Example 5·5 shows that non-trivial solvability of (26) over ${\mathbf{Q}}$ may not guarantee the existence of unramified extensions $K\subset F(x,y,z)$ , whereas the slightly stronger conditions of Lemma 4.2 do. More precisely, by the classical local-global principle for conics, assuming solvability of (26) amounts to having quadratic Hilbert symbols $(d_1,d_2)_p=1$ for all finite primes p, with $(d_1,d_2)_\infty=1$ then being implied by the product formula. At $p\nmid D=d_1d_2$ , including $p=2$ , we have $(d_1,d_2)_p=1$ . For an odd prime p dividing $D=d_1d_2$ , say $d_1$ , the Hilbert symbol condition at p is

\begin{equation*}(d_1,d_2)_p={\left(\frac{d_2}{p}\right)}=1,\end{equation*}

exactly as in condition (iv) of Lemma 4.2. For $p=2$ dividing $d_1$ however, we have $d_2\equiv1\bmod4$ and obtain

(27) \begin{equation}(d_1,d_2)_2={\left(\frac{d_2}{2}\right)}^{ord_2(d_1)}=1.\label{eqn27}\end{equation}

If D has 2-part $t_D=-4$ in (5) and we have $d_2\equiv 5\bmod 8$ , such as for $D=-820$ and $d_2=205$ above, condition (27) does not imply condition (iv) of Lemma 4.2 for $p=2$ , as we have $(d_1,d_2)_2=1$ but $({d_2}/{2})=-1$ . This discrepancy will lead us to the notion of 2-minimal ramification in Definition 7·5.

6. Discovering Redei reciprocity

Explicit computations of Rédei symbols exhibit a ‘reciprocity law’ that we can discover already by looking at the most classical example

\begin{equation*}K={\mathbf{Q}}(\sqrt {-p}),\end{equation*}

with p an odd prime. Having $r_2=t-1=1$ for K amounts to having $p\equiv 1\bmod 4$ and $D=-4p$ by Theorem 2·1; otherwise C has odd order. Assuming this, $R_4$ in Theorem 3·1 has rank 0 if and only if we have $(\tfrac{p}{2})=(\tfrac{2}{p})=1$ , so $r_4=1$ happens for $p\equiv 1\bmod 8$ . We further assume $p\equiv 1\bmod 8$ .

As $(\sqrt{-p})|p$ is principal, the class of the non-principal prime ${\mathfrak{p}}_2|2$ generates $C[2]=C[2]\cap 2C$ , and $R_8$ consists of a single Rédei symbol $[-4,p,2]$ . The case $r_8=1$ in which the symbol vanishes occurs when the prime ${\mathfrak{p}}_2$ of K splits completely in the 4-Hilbert class field $H_4(-p)$ of K. Solving

\begin{equation*}x^2+4y^2-pz^2=0\end{equation*}

with $z=1$ , we can generate $H_4(-p)$ over $E={\mathbf{Q}}(i, \sqrt p)$ by adjoining a square root of $\pi=x+2iy$ . Now ${\mathfrak{p}}_2$ splits into 4 primes in the extension $K\subset H_4(-p)$ if and only if the prime $(1+i)$ over 2 in ${\mathbf{Q}}(i)$ splits into 4 primes in the extension

\begin{equation*}{\mathbf{Q}}(i)\subset H_4(-p)={\mathbf{Q}}(\sqrt{-p}, i, \sqrt\pi)={\mathbf{Q}}(i, \sqrt \pi,\sqrt{\bar\pi}).\end{equation*}

This shows that $[-4,p,2]$ can be viewed as a ‘Kronecker symbol’ $(\tfrac{\pi}{1+i})$ in ${\mathbf{Q}}(i)$ . By class field theory (or quadratic reciprocity) over ${\mathbf{Q}}(i)$ , this symbol is simply the Legendre symbol $(\tfrac{1+i}{p})$ , which is well defined for $p\equiv 1\bmod 8$ , and we have

(28) \begin{equation}r_8=1\quad \Longleftrightarrow\quad\text{{p} splits completely in ${\mathbf{Q}}(\zeta_8,\sqrt{1+i})$.}\label{eqn28}\end{equation}

We deduce that the prime ${\mathfrak{p}}_2$ over 2 splits completely in the unramified extension $K\subset H_4(-p)$ if and only if p splits completely in ${\mathbf{Q}}(\zeta_8,\sqrt{1+i})$ . By the case $(a,b)=(-1,2)$ of Lemma 5·1, the field ${\mathbf{Q}}(\zeta_8,\sqrt{1+i})= {\mathbf{Q}}(i, \sqrt 2,\sqrt{1+i})$ is dihedral over ${\mathbf{Q}}$ , just like $H_4(-p)$ . In fact, both fields are abelian of exponent 2 over ${\mathbf{Q}}(i)$ , quadratic over respectively ${\mathbf{Q}}(i,\sqrt 2)$ and ${\mathbf{Q}}(i, \sqrt p)$ , and cyclic over respectively ${\mathbf{Q}}(\sqrt {-2})$ and ${\mathbf{Q}}(\sqrt{-p})$ . We have proved a special case of Rédei reciprocity, and in terms of Rédei symbols it can be suggestively formulated as

(29) \begin{equation}[-4,p,2]=[-4,2,p].\label{eqn29}\end{equation}

The symbol on the left is defined by Definition 4·4 for symbols $[d_1,d_2,m]$ , but the symbol on the right is not. It is natural to take $d_1$ and $d_2$ up to squares, yielding the formulation $[-1,p,2]=[-1,2,p]$ , but the symbol $[-1,2,p]$ refers to the splitting of the primes over p in

\begin{equation*}{\mathbf{Q}}(\sqrt{-2})\subset E={\mathbf{Q}}(i,\sqrt 2)\subset F=E(\sqrt{1+i}),\end{equation*}

a cyclic quartic extension that is totally ramified over 2. Primes $p\equiv 1 \bmod 8$ are totally split in E, and split or inert in $E\subset F$ depending on the value of $[-1,2,p]$ .

As we can swap the arguments $-1$ and p in the symbol $[-1,p,2]$ by its very definition, and 2 and p by what we just proved, one naturally wonders whether it also equals a symbol $[2,p,-1]$ that describes the splitting of “ $-1$ ” in the narrow 4-Hilbert class field $H_4(2p)$ of ${\mathbf{Q}}(\sqrt{2p})$ . By Theorem 3·1, the field $H_4(2p)$ is quadratic over the totally real field ${\mathbf{Q}}(\sqrt 2,\sqrt p)$ for $p\equiv1\bmod 8$ . Now Frobenius symbols at “ $-1$ ”, which over ${\mathbf{Q}}$ raise roots of unity to the power $-1$ , arise in class field theory as complex conjugations, and act trivially on totally real fields. The dihedral field $H_4(2p)$ is abelian of exponent 2 over ${\mathbf{Q}}(\sqrt 2)$ , and it is totally real if and only if its conductor over ${\mathbf{Q}}(\sqrt 2)$ is p, not $p\cdot\infty$ . Looking at the ray class group

\begin{equation*}({\mathbf{Z}}[\sqrt2]/p{\mathbf{Z}}[\sqrt2])^*/\langle -1, 1+\sqrt 2\rangle\end{equation*}

modulo p of ${\mathbf{Q}}(\sqrt 2)$ , we see that $H_4(2p)$ is real exactly when the fundamental unit $1+\sqrt 2\in {\mathbf{Q}}(\sqrt 2)$ is a square modulo p, and this happens for the primes that split completely in the dihedral field ${\mathbf{Q}}(\zeta_8, \sqrt{1+\sqrt 2})$ . By equation (24) for $(a,b)=(-1,2)$ and $(x,y,z)=(1,1,1)$ , this is the same field as ${\mathbf{Q}}(\zeta_8,\sqrt{1+i})$ , so the Rédei symbol

(30) \begin{equation}[-1,p,2]=[-1,2,p]=[2,p,-1],\label{eqn30}\end{equation}

when properly defined, is invariant under all permutations of its arguments.

In Rédei’s own definition, $[-1,2,p]$ does not exist, and $[p,2,-1]$ is trivial for all p. Our definition in the next section introduces a notion of minimal ramification for extensions $K\subset F$ as in (25), correcting the definition found in [Reference Corsman5].

7. Redei symbols

In order to obtain Rédei reciprocity, we generalize the symbol $[d_1, d_2, m]$ in Definition 4·4 beyond the setting of dihedral fields F containing ${\mathbf{Q}}(\sqrt{d_1},\sqrt{d_2})$ that are cyclic and unramified over $K={\mathbf{Q}}(\sqrt{d_1d_2})$ and norms m of ambiguous ideals ${\mathfrak{m}}$ of K with trivial Artin symbol in the genus field of K. As $d_1$ and $d_2$ encode quadratic fields, and m is the norm of an ideal with ideal class in a subgroup of C of exponent 2, the general Rédei symbol [a,b,c] naturally takes its arguments in the group ${\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ . It will be linear in each of its arguments.

Every $\bar a\in{\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ is uniquely represented by a squarefree integer a, and corresponds to a number field ${\mathbf{Q}}(\sqrt a)$ that is quadratic for $a\ne 1$ . Given non-trivial elements in ${\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ represented by squarefree integers a, b, the extension

(31) \begin{equation}{\mathbf{Q}}(\sqrt{ab})=K\subset E={\mathbf{Q}}(\sqrt a, \sqrt b)\label{eqn31}\end{equation}

is quadratic, and ${\mathbf{Q}}\subset E$ is unramified at primes outside the discriminants $\Delta(a)$ and $\Delta(b)$ of the quadratic fields corresponding to a and b.

We now assume that a and b have relative quadratic Hilbert symbols $(a,b)_p=1$ for all primes p. As we observed at the end of Section 5, this amounts to saying that the equation

(32) \begin{equation}x^2-a y^2-b z^2=0\label{eqn32}\end{equation}

admits non-zero rational solutions. By Corollaries 5.2 and 5.3, such a solution (x,y,z) generates a cyclic quartic extension

(33) \begin{equation}{\mathbf{Q}}(\sqrt{ab})=K\subset F=F(x,y,z)=E(\sqrt \beta)=E(\sqrt\alpha)\label{eqn33}\end{equation}

in which we take $\beta=x+y\sqrt a$ and $\alpha=2(x+z\sqrt b)$ . The field F is dihedral over ${\mathbf{Q}}$ for $a\ne b$ , and cyclic over ${\mathbf{Q}}(\sqrt{ab})={\mathbf{Q}}$ for $a=b$ . It is uniquely determined by a and b up to twisting by rational quadratic characters $\chi_t$ , with $t\in{\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ . In fact, the asymmetry in (33) in the definition of $\alpha$ and $\beta$ coming out of (32) can be seen as making a somewhat arbitrary choice between F and $F_2$ . Here we use the twisting notation

(34) \begin{equation}F_t=F(tx,ty,tz)=E(\sqrt {t\beta})=E(\sqrt{t\alpha})\label{eqn34}\end{equation}

for $F=F(x,y,z)$ from Section 5.

Before defining the general symbol [a,b,c], we start with the special case of a squarefree integer $a=b\ne1$ . Then $K\subset E$ in (31) is the extension ${\mathbf{Q}}\subset {\mathbf{Q}}(\sqrt a)$ . As Hilbert symbols satisfy $(a,-a)_p=1$ for all $a\in{\mathbf{Q}}^*$ , our assumption on a is $(a,-1)_p=1$ for all p. By (18), this means that the discriminant $\Delta(a)\in\{a,4a\}$ is positive and without prime divisors $3\bmod4$ , i.e., in the set ${\mathcal{D}}$ of Conjecture 3·2. For such a, we can write the associated Dirichlet character $\chi_a:G_{\mathbf{Q}}\to{\mathbf{F}}_2$ in the notation of (14) as

\begin{equation*}\chi_a=\sum_{p|a} \chi_p,\end{equation*}

with $\chi_2=\chi_8$ the quadratic character associated to ${\mathbf{Q}}(\sqrt 2)$ . Let $\psi_p$ be a character of order 4 modulo $p\equiv1\bmod 4$ , and $\psi_2$ a character of order 4 on $({\mathbf{Z}}/16{\mathbf{Z}})^*$ . Then

(35) \begin{equation}\psi_a=\sum_{p|a} \psi_p\label{eqn35}\end{equation}

is a quartic Dirichlet character of conductor a (or 8a when a is even) that satisfies $2\psi_a=\chi_a$ . It corresponds to a cyclic quartic field $F_{a,a}$ containing $E={\mathbf{Q}}(\sqrt a)$ . Cyclic quartic ${\mathbf{Q}}\subset F$ containing E are unique up to quadratic twists, as $2\psi_F=\chi_a$ only defines $\psi_F$ up to a quadratic character. Clearly $E\subset F$ will be ramified at all primes dividing a, but not at other primes if we take $\psi_F=\psi_a$ as in (35). Such an extension ${\mathbf{Q}}\subset F_{a,a}$ of minimal ramification is not unique, as we can add $\chi_p$ to each $\psi_p$ of odd conductor p in (35), and $\chi_2$ and $\chi_{-1}$ to $\psi_2$ . This makes $\psi_a$ unique up to twisting by sums of characters $\chi_p$ with $p|a$ , and in addition the character $\chi_{-1}$ in case a is even. For $a=1$ we define $\psi_a$ to be the trivial character.

In terms of Corollary 5·3, a minimally ramified cyclic quartic extension ${\mathbf{Q}}\subset F_{a,a}$ containing $E={\mathbf{Q}}(\sqrt a)$ is unique up to twisting by t in the finite twisting subgroup

(36) \begin{equation}T_{a,a}\subset {\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2\label{eqn36}\end{equation}

generated by the residue classes of the odd primes $p=p^*$ dividing a and, for a even, of $-1$ and 2. It follows that for squarefree integers a and c satisfying

(37) \begin{equation}\gcd(\Delta(a),\Delta(c))=1\quad\hbox{and}\quad\hbox{$(a,a)_p=(a,c)_p=1$ for all {p},}\label{eqn37}\end{equation}

we have $\chi_t(c)=0$ for $t\in T_{a,a}$ , and a well defined biquadratic character

(38) \begin{equation}\psi_a(c)={\left(\frac{c}{a}\right)}_4\in {\mathbf{F}}_2.\label{eqn38}\end{equation}

In terms of Artin symbols in the cyclotomic field ${\mathbf{Q}}(\zeta_a)$ (or ${\mathbf{Q}}(\zeta_{8a})$ ) containing $F_{a,a}$ , we have

(39) \begin{equation}\psi_a(c)= Art(c, F_{a,a}/{\mathbf{Q}})\in Gal(F_{a,a}/E)={\mathbf{F}}_2.\label{eqn39}\end{equation}

Note that a and $\Delta(a)\in{\mathcal{D}}$ have the same prime factors by the hypothesis $(a,a)_p=1$ for all p. Thus (37) implies that for a even, with $-1, 2\in T_{a,a}$ , we have $c\equiv 1\bmod 4$ by the gcd-condition, and then $c\equiv 1\bmod 8$ by the condition $(a,c)_2=1$ , yielding $\chi_2(c)=\chi_{-1}(c)=0$ . For odd primes p dividing a, with $p\in T_{a,a}$ , the conditions $(a,c)_p=1$ and $\chi_p(c)=0$ coincide.

Viewing $\psi_a$ in (38) as defined on the subgroup of ${\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ generated by the integers c satisfying (37), we obtain the following special Rédei symbol. For $a, c \in {\mathbf{Q}}^*/{{\mathbf{Q}}^*}^2$ satisfying (37), we take (38) to define

\begin{equation*}[a,a,c]=\psi_a(c)={\left(\frac{c}{a}\right)}_4\in {\mathbf{F}}_2.\end{equation*}

Definition 7·1. For the general case, we take $a, b\in{\mathbf{Z}}_{\ne1}$ to be different squarefree integers, assuming

(40) \begin{equation}(a,b)_p=1\quad\hbox{for all primes {p}}\label{eqn40}\end{equation}

in order to have non-trivial solvability of (32). Then the quadratic extension

\begin{equation*}{\mathbf{Q}}(\sqrt{ab})=K\subset E={\mathbf{Q}}(\sqrt a, \sqrt b)\end{equation*}

from (31) is only ramified at primes ${\mathfrak{p}} | \gcd(\Delta(a),\Delta(b))$ , and such ${\mathfrak{p}}$ will be totally ramified in every cyclic quartic extension $K\subset F$ in (33). Generalizing (39), we are to define the Rédei symbol in Definition 7·8 as

\begin{equation*}[a,b,c]=Art_c (F_{a,b}/K)\in Gal(F_{a,b}/E)={\mathbf{F}}_2,\end{equation*}

the Artin symbol of an ideal ${\mathfrak{c}}$ in K corresponding to c in a cyclic quartic extension $K\subset F_{a,b}$ constructed from a solution to (32) that is minimally ramified over E.

For odd primes $p\nmid \gcd(\Delta(a),\Delta(b))$ , one can avoid ramification over p in $E\subset F$ by passing, if needed, to the quadratic twist $F_{p^*}$ from (34), with $p^*=\pm p$ as in (5).

Proposition 7·2. Let $a,b\in{\mathbf{Z}}_{\ne1}$ be distinct and squarefree, $p\nmid\Delta(b)$ an odd prime, and $K={\mathbf{Q}}(\sqrt{ab})\subset F=E(\sqrt\beta)$ as in (33).

  1. (i) If p divides $\Delta(a)$ , then $K\subset F$ is unramified over p.

  2. (ii) If p does not divide $\Delta(a)$ , then exactly one of $K\subset F$ and $K\subset F_{p^*}$ is unramified over p.

Proof. Consider F as a quartic extension of $K_a={\mathbf{Q}}(\sqrt a)$ . The intermediate field $E=K(\sqrt b)=K_a(\sqrt b)$ is a quadratic extension of both K and $K_a$ that is unramified at primes dividing p, as we have $p\nmid \Delta(b)$ . It follows that $K\subset F$ is unramified over p if and only if $K_a\subset F$ is.

Write $F=K_a(\sqrt \beta, \sqrt{\beta'})$ , with $\beta\in K_a$ of norm $\beta\beta'=x^2-ay^2=bz^2$ . As p is odd, $K_a\subset F$ is unramified over p if and only if $\beta$ and $\beta'$ have even valuation at the primes ${\mathfrak{p}}|p$ in $K_a$ . For a prime ${\mathfrak{p}}|p$ of ramification index $e_{{\mathfrak{p}}/p}$ in $K_a$ , we have

\begin{equation*}ord_{\mathfrak{p}}(\beta\beta')=e_{{\mathfrak{p}}/p}ord_p(bz^2)=2e_{{\mathfrak{p}}/p}ord_p(z).\end{equation*}

In the ramified case $p|\Delta(a)$ , we have $e_{{\mathfrak{p}}/p}=2$ and $ord_{\mathfrak{p}}(\beta\beta')=2ord_{\mathfrak{p}}(\beta)\equiv 0\bmod 4$ , proving (i). In the unramified case $p\nmid\Delta(a)$ , we have $e_{{\mathfrak{p}}/p}=1$ and $ord_{\mathfrak{p}}(\beta)\equiv ord_{\mathfrak{p}}(\beta')\bmod 2$ . Moreover, we have $ord_{\mathfrak{p}}(p^*\beta)=ord_{\mathfrak{p}}(\beta)+1$ , so ${\mathfrak{p}}$ is unramified in exactly one of F and $F_{p^*}$ , proving (ii).

As twisting by $p^*$ does not change ramification outside p, Proposition 7·2 shows that $K\subset F$ in (33) can be chosen to be unramified at all odd $p\nmid \gcd(\Delta(a),\Delta(b))$ . For $p=2$ we can twist by $-1$ and 2, with the following outcome.

Proposition 7·3. Let $a,b\in{\mathbf{Z}}_{\ne1}$ be distinct and squarefree, $b=\Delta(b)\equiv1\bmod 4$ , and $K={\mathbf{Q}}(\sqrt{ab})\subset F=E(\sqrt \beta)$ as in (33).

  1. (i) If $\Delta(a)$ is odd, then ${\mathbf{Q}}\subset F_t$ is unramified at 2 for a unique $t\in\{\pm1,\pm2\}$ .

  2. (ii) If $\Delta(a)$ is even and $\Delta(b)$ is $1\bmod 8$ , then $K\subset F_t$ is unramified over 2 for exactly two values of $t\in\{\pm1,\pm2\}$ .

  3. (iii) If $\Delta(a)$ is even and $\Delta(b)$ is $5\bmod 8$ , then $K\subset F_t$ is ramified over 2 for all $t\in{\mathbf{Q}}^*$ , and $\Delta(a)$ is $4\bmod 8$ .

Proof. Just as for odd p, we consider the extension $K_a\subset F=K_a(\sqrt \beta, \sqrt{\beta'})$ . As before, up to squares, $\beta$ is a 2-unit in $K_a$ if $\Delta(a)$ is even, and exactly one of $\beta$ and $2\beta$ is a 2-unit $K_a$ if $\Delta(a)$ is odd. However, for a 2-unit to have a square root that is unramified at 2, we need the stronger condition that it is a square modulo 4.

We can assume, possibly after twisting F by $t=2$ , that $\beta$ is a 2-unit in the ring of integers ${\mathcal{O}}$ of $K_a$ . For $2\nmid \Delta(a)$ , the group $({\mathcal{O}}/4{\mathcal{O}})^*$ has order 4 or 12, depending on whether 2 is split or inert in ${\mathcal{O}}$ , and the squares form a subgroup of index 4. Together with $-1$ , they generate the kernel of the surjective norm map

\begin{equation*}N: ({\mathcal{O}}/4{\mathcal{O}})^*\longrightarrow ({\mathbf{Z}}/4{\mathbf{Z}})^*.\end{equation*}

By assumption, we have $\beta\beta'\equiv b\equiv1\bmod 4{\mathcal{O}}$ , so the residue classes $\beta, \beta'\in\ker N$ are squares in $({\mathcal{O}}/4{\mathcal{O}})^*$ for a unique ‘sign choice’ of $\beta$ , and ${\mathbf{Q}}\subset F_t$ is unramified at 2 for a unique value $t\in\{\pm1,\pm2\}$ . This proves (i).

For $2|\Delta(a)$ , the group $({\mathcal{O}}/4{\mathcal{O}})^*=({\mathcal{O}}/{\mathfrak{p}}^4{\mathcal{O}})^*$ has order 8, and its subgroup of squares, of index 4, is of order 2. The norm ${\mathcal{O}}={\mathbf{Z}}[\sqrt a]\to {\mathbf{Z}}$ induces a map

\begin{equation*}N: ({\mathcal{O}}/4{\mathcal{O}})^*\longrightarrow ({\mathbf{Z}}/8{\mathbf{Z}})^*\end{equation*}

for which the image, of order 2, is generated by $1-a\bmod 8$ when $a\equiv\pm2\bmod 8$ is even, and by $5\bmod 8$ when $a\equiv -1\bmod 4$ is odd.

In the case where a is even, $\ker N$ is non-cyclic of order 4, generated by $-1$ and the squares in $({\mathcal{O}}/4{\mathcal{O}})^*$ , and it contains $\beta\bmod 4{\mathcal{O}}$ as $\beta\beta'\equiv b\bmod 8$ is not $5\bmod 8\notin im N$ . In this case, we have $\Delta(b)= b\equiv1\bmod 8$ , and we conclude just as before that exactly one of F and $F_{-1}$ is unramified over K at 2. By the same argument applied to $F_2$ , one of $F_2$ and $F_{-2}$ is unramified over K at 2, so $K\subset F_t$ is unramified over 2 for exactly two values $t\in\{\pm1, \pm2\}$ , as stated in (ii).

In the remaining case $a\equiv -1\bmod 4$ , or $\Delta(a)\equiv 4\bmod 8$ , the residue class of

(41) \begin{equation}\tau=(1+\sqrt a)^2/2=(1+a)/2+\sqrt a\label{eqn41}\end{equation}

in $({\mathcal{O}}/4{\mathcal{O}})^*$ , which equals $\sqrt a\bmod4{\mathcal{O}}$ for $a\equiv -1\bmod 8$ and $2+\sqrt a\bmod4{\mathcal{O}}$ for $a\equiv 3\bmod 8$ , has square $-1\bmod4{\mathcal{O}}$ , so it is of order 4 and generates $\ker N$ .

We now have 2 cases. For $\Delta(b)= b\equiv1\bmod 8$ we have $\beta\bmod 4{\mathcal{O}}\in\ker N$ , and twisting by