Caliber number 2 for real quadratic fields with radicand $d\equiv 5\pmod 8$: completing the Jun–Lee classification

The caliber number $\kappa(d)$ of a real quadratic field $K=\mathbb{Q}(\sqrt d)$ is the number of reduced primitive binary quadratic forms of its discriminant, equivalently the sum over ideal classes of the periods of the associated continued fractions; since the class number satisfies $h(d)\le\kappa(d)$, small caliber is an unconditional analogue of small class number. Jun and Lee (J. Number Theory 130, 2010) determined all fields of caliber 1 and all fields of caliber 2 with $d\not\equiv 5\pmod 8$, unconditionally and without any assumption on $\zeta_K(1/2)$, but their reduction of the caliber-2 problem to class-number-one statements breaks down for $d\equiv 5\pmod 8$, where the prime 2 is inert, and this residue class was left open. The problem is to complete that classification: determine, unconditionally, all real quadratic fields $\mathbb{Q}(\sqrt d)$ with $d$ squarefree, $d\equiv 5\pmod 8$, and $\kappa(d)=2$.

Background

For a squarefree positive integer $d$ (the radicand), $K=\mathbb{Q}(\sqrt d)$ is a real quadratic field; its ring of integers $\mathcal{O}_K$ has discriminant $D=d$ if $d\equiv 1\pmod 4$ and $D=4d$ otherwise. A primitive binary quadratic form $[A,B,C]=AX^2+BXY+CY^2$ with integer coefficients (primitive meaning $\gcd(A,B,C)=1$) has discriminant $B^2-4AC$; the group $GL_2(\mathbb{Z})$ acts on the forms of a fixed discriminant and the number of orbits is the class number $h(d)$. Following Gauss and Dickson, a form is called reduced if $A>0$, $B<0$, $C<0$, $|B|<\sqrt D$ and $\sqrt D-|B|<2A<\sqrt D+|B|$. Lachaud (1987) defined the caliber number $\kappa(d)$ as the number of reduced primitive forms of discriminant $D$; equivalently $\kappa(d)=\sum_{[Q]} m([Q])$, where $m([Q])$ is the length of the period of the continued fraction expansion of a reduced quadratic irrationality in the class $[Q]$, i.e. the total number of reduced forms summed over the period cycles of all classes (this cycle structure is known as the infrastructure of $K$). Each class contains at least one reduced form, so $h(d)\le\kappa(d)$: small caliber forces small class number. This makes $\kappa$ a real-quadratic counterpart of the class number in the spirit of the Gauss class number problem, which for imaginary quadratic fields ends in finite lists (exactly nine fields of class number one) while for real quadratic fields infinitely many class-number-one fields are believed to exist and no unconditional finiteness method is known. Lachaud, assuming $\zeta_K(1/2)\le 0$ for the Dedekind zeta function $\zeta_K$ of $K$ (the analogue of the Riemann zeta function for the integers of $K$; a condition implied on the interval $(0,1)$ by the generalized Riemann hypothesis, GRH), proved the lower bound $\kappa(d)>\frac{1}{8.46}\log(d-3)$ and determined the caliber-1 fields. Jun and Lee made such results unconditional: they proved $\kappa(d)=1$ exactly for $d\in\{2,5,13,29,53,173,293\}$ (their printed list omits $d=5$, which has the single reduced form $[1,-1,-1]$; this is a typographical slip, consistent with Biró's theorem and with later tables), gave the elementary lower bound $\kappa(d)>2\big[\frac{\log(\sqrt D/2)}{\log p}\big]$ whenever a rational prime $p$ splits in $K$ (i.e. $p\mathcal{O}_K$ is a product of two distinct prime ideals), and determined $\kappa(d)=2$ for $d\not\equiv 5\pmod 8$: exactly $d\in\{3,6,11,38,83,227\}$. Their method reduces the caliber-2 question to class-number statements for Richaud–Degert (R-D) fields, those with $d=n^2+r$ and $r\mid 4n$ (narrow when $|r|\in\{1,4\}$, wide otherwise), the family whose fundamental unit (a generator of the unit group of $\mathcal{O}_K$ up to sign) is explicitly computable; the key inputs are Biró's unconditional resolution of Yokoi's conjecture ($d=n^2+4$: $h=1$ exactly for $d\in\{5,13,29,53,173,293\}$) and Chowla's conjecture ($d=n^2+1$), the class-number criteria of Byeon–Kim, and Lee's list for $d=n^2\pm 2$. For $d\equiv 5\pmod 8$ the method breaks: then $D=d\equiv 5\pmod 8$ and the prime 2 is inert in $K$, i.e. the ideal $2\mathcal{O}_K$ remains prime, so the lemmas about ideals above 2 are unavailable, and the case $\kappa(d)=2$ with $d\equiv 5\pmod 8$ was explicitly left open. Later unconditional class-number-one results cover the relevant R-D subfamilies: Byeon, Kim and Lee (2007, 'Mollin's conjecture') settled $d=n^2-4$ ($h=1$ exactly for $d\in\{5,21,77,437\}$ with $n$ odd), Lee (2009) settled wide R-D types with $d\not\equiv 5\pmod 8$, and Biró–Lapkova (2016) settled $d=(an)^2+4a$ for odd $a,n$ ($h=1$ forces $d\le 1253$). On the class-number-two side, Byeon–Lee (2008) settled $d=n^2+1$ ($h=2$ exactly for $d\in\{10,26,122,362\}$), Lee (2009) settled narrow R-D types with $d\not\equiv 5\pmod 8$, Mollin–Williams (1991) and Mollin (2009) classified R-D fields of class number 2 completely except for possible exceptions removable only under GRH, and Kopp–Lagarias (2026) classified unit-generated orders $\mathbb{Z}[\varepsilon]$ (discriminants $n^2\pm 4$) of class number one and computed 2-torsion class-group lists for $\Delta<10^{10}$ that are conjectural beyond. The adjacent caliber literature (Kaneko–Mori; Kaneko–Sakata–Takeuchi; Fujisawa) studies congruences and parity of calibers, not the caliber-2 census.

Problem Statement

Determine, unconditionally and completely, all real quadratic fields $K=\mathbb{Q}(\sqrt d)$ with $d$ a positive squarefree integer satisfying $d\equiv 5\pmod 8$, whose caliber number $\kappa(d)$ equals 2. Here $d\equiv 1\pmod 4$, so the field discriminant is $D=d$, and $\kappa(d)$ is defined as the number of reduced primitive binary quadratic forms $[A,B,C]=AX^2+BXY+CY^2$ of discriminant $B^2-4AC=D$ with integer coefficients, where primitive means $\gcd(A,B,C)=1$ and reduced means $A>0$, $B<0$, $C<0$, $|B|<\sqrt D$, and $\sqrt D-|B|<2A<\sqrt D+|B|$. This completes the classification of Jun and Lee, who settled the complementary residue classes ($d\not\equiv 5\pmod 8$: exactly $d\in\{3,6,11,38,83,227\}$). Equivalently, the target is the finite list $L$ of squarefree radicands $d\equiv 5\pmod 8$ with $\kappa(d)=2$: every $d\in L$ has $\kappa(d)=2$, and no squarefree $d\equiv 5\pmod 8$ outside $L$ has $\kappa(d)=2$. The classification is to be unconditional: it assumes nothing about the Dedekind zeta function $\zeta_K$ of $K$ (such as $\zeta_K(1/2)\le 0$) and nothing about the generalized Riemann hypothesis (GRH).

References

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