Unproved arithmetic of Broadhurst's level-48 class invariants at discriminants N ≡ 3 (mod 8): the Chowla–Selberg unit λ and mod-64-signature generation of the Hilbert class field

Broadhurst (arXiv:0807.2976, 2008) reduced the complete elliptic integral $K_N$ and the singular modulus $k_N$ at square-free discriminants $N \equiv 3 \pmod 8$ to radicals and $\Gamma$-values by means of two new level-48 class invariants $f$, $g$, built from Weber's eta-quotient value $r$ through nested radicals whose three signs (the signature $[S_1,S_2,S_3]$) are fixed by the residue of $N$ modulo 64, together with the Chowla–Selberg factor $\lambda$. Every arithmetic property on which this construction rests was left conjectural: that $\lambda$ is a unit of the Hilbert class field for prime $N \equiv 3 \pmod 4$; that $f$ and $g$ are algebraic integers whose minimal polynomials generate the Hilbert class field (at least one of $f, g$ for every such $N$, both for $N > 1099$) and, for $N > 1099$, have degree $h(-N)$; why the signature and the asymptotic prefactor $\alpha(N) = \sqrt{2}\cos(k\pi/16)$ of $g$ depend only on $N \bmod 64$; and when $\lambda$ rather than $\lambda^2$ is a unit at composite discriminants.

Background

This problem sits in explicit class field theory for imaginary quadratic fields, as it enters the exact evaluation of complete elliptic integrals.

For a positive square-free integer $N$, the $N$-th singular value (or singular modulus) $k_N \in [0,1]$ is the algebraic number defined by $\mathrm{AGM}(1,\sqrt{1-k_N^2}) = \sqrt{N}\,\mathrm{AGM}(1,k_N)$, where $\mathrm{AGM}(a,b)$ is the arithmetic–geometric mean, obtained by iterating $(a,b) \mapsto ((a+b)/2, \sqrt{ab})$. The associated complete elliptic integral $K_N = \int_0^1 \mathrm{d}x/\sqrt{(1-x^2)(1-k_N^2 x^2)} = (\pi/2)/\mathrm{AGM}(1,\sqrt{1-k_N^2})$ is a period of an elliptic curve with complex multiplication (CM) — meaning that the curve's endomorphism ring is an order in the imaginary quadratic field $K = \mathbb{Q}(\sqrt{-N})$ rather than just $\mathbb{Z}$. CM theory makes $k_N$ and related quantities algebraic: they lie in the Hilbert class field $H_N$, the maximal unramified abelian extension of $K$, whose degree over $K$ is the class number $h(-N)$ — the size of the ideal class group, equivalently of the group $H$ of classes of primitive binary quadratic forms $ax^2+bxy+cy^2$ of discriminant $b^2-4ac=-N$ under Gauss composition.

The Dedekind eta function $\eta(z) = e^{\pi i z/12}\prod_{n \ge 1}(1-e^{2\pi i n z})$, a holomorphic function on the upper half-plane transforming with weight $1/2$ under $SL_2(\mathbb{Z})$, is the building block of the classical class invariants: values of eta quotients at CM points $z = (b+\sqrt{-N})/(2a)$, such as the Weber functions, whose minimal polynomials over $\mathbb{Q}$ have degree $h(-N)$ and generate $H_N$ far more economically than the $j$-invariant (Schertz 2002; Enge–Schertz 2004). A modular function 'of level $n$' is one invariant under a congruence subgroup of level $n$; smaller level generally means smaller height of the resulting class polynomial. The standard tool for proving that such a value is a class invariant is Shimura reciprocity, the theorem describing the Galois action on CM values of modular functions of level $n$ through the ray class group of conductor $n$.

At discriminants $N \equiv 3 \pmod 8$ with $\gcd(N,3)=1$, the Weber-type value $r = e^{-\pi i/24}\,\eta((1+\sqrt{-N})/2)/\eta(\sqrt{-N})$ has minimal polynomial of degree $3h(-N)$ rather than $h(-N)$ — the classical 'trebling' obstruction, which made this residue class unattractive from Weber's time until it was set aside in early elliptic-curve primality proving. Broadhurst (2008) overcame it empirically: with a signature $[S_1,S_2,S_3]$ of three signs determined by $N \bmod 64$ through an eight-line table (Eq. (23) of the source; e.g. $[+1,-1,-1]$ for $N \equiv 3$, $[-1,-1,-1]$ for $N \equiv 35$, $[+1,+1,+1]$ for $N \equiv 27 \pmod{64}$), he formed $s = S_1\Big(1 + S_2\Big(\tfrac{1}{2} + S_3\big(\tfrac{1}{8} - r^{-12}\big)^{1/2}\Big)^{1/2}\Big)^{1/2}, \qquad f = \frac{r}{2} - \frac{s}{\sqrt{r}}, \qquad g = -\frac{1}{r} + s\sqrt{r},$ and found that $f$ and $g$ appear to be algebraic integers generating $H_N$ — both, with minimal polynomials of degree $h(-N)$, for $N > 1099$ — so that $k_N^2 = \tfrac{1}{2}-\sqrt{\tfrac{1}{4}-\tfrac{16}{r^{24}}}$ can be recovered from degree-$h$ rather than degree-$3h$ polynomials. The invariant $g$ is a level-48 modular function with the slow growth $g = \alpha(N)\,e^{\pi\sqrt{N}/48} + o(1)$, where the prefactor $\alpha(N) = \sqrt{2}\cos(k\pi/16)$ ($k$ odd, $1 \le k \le 15$) is again fixed by $N \bmod 64$; this halving of the exponential rate is the source of the observed economy — for $N = 2317723$ the height of the minimal polynomial of $g$ is less than the cube root of the height of the polynomial produced by the double eta-quotient method underlying PARI/GP's quadhilbert.

The construction also interacts with the evaluation of $K_N$ itself. For square-free $N \equiv 3 \pmod 4$, $N > 3$, the eta form of the Chowla–Selberg formula (Selberg–Chowla 1967), $G_N = \prod_{k=1}^{N}\Gamma(k/N)^{(\frac{-N}{k})} = (2\pi N)^{h}\prod_{[a,b,c] \in H}\frac{1}{a}\Big|\eta\Big(\frac{b+\sqrt{-N}}{2a}\Big)\Big|^{4}$ (with $(\frac{-N}{k})$ the Kronecker symbol), reduces $K_N = (r/2)^2\sqrt{(2\pi/N)(\lambda^4 G_N)^{1/h}}$ to $\Gamma$-values and the algebraic Chowla–Selberg factor $\lambda = \prod_{[a,b,c] \in H} a^{1/4}\Big|\frac{\eta((1+\sqrt{-N})/2)}{\eta((b+\sqrt{-N})/(2a))}\Big|.$ Broadhurst conjectured that $\lambda$ is a unit (an invertible algebraic integer, i.e. of norm $\pm 1$) of the Hilbert class field $H_N$.

All of this was established numerically — degrees of the minimal polynomials of $f$ and $g$ checked for $1099 < N < 100000$, generated fields compared with quadhilbert output for all square-free $N \equiv 3 \pmod 8$ coprime to 3 below 3500 and for primes below 6000, unit status of $\lambda$ checked for the 155 primes $N < 2000$ at 15000 digits of precision, and the case $N = 2317723$ at 40000 digits — and left unproved. Even the mod-64 rule was discovered empirically. Those conjectures are the problem below.

Problem Statement

Prove, or refute by explicit counterexample, the conjectures on which Broadhurst's 2008 evaluation of the complete elliptic integral $K_N$ and the singular modulus $k_N$ at discriminants $N \equiv 3 \pmod 8$ rests. Throughout, $K = \mathbb{Q}(\sqrt{-N})$, $H_N$ is its Hilbert class field, $h = h(-N)$ is the class number, $H$ is the group of classes of primitive binary quadratic forms of discriminant $-N$, and $\lambda$, $r$, $s$, $f$, $g$ are as defined in the Background, with the signature $[S_1,S_2,S_3]$ fixed by $N \bmod 64$ through the eight-line table. The parts, which jointly give the construction a rigorous arithmetic foundation, are:

(a) Unit conjecture (Conjecture 1 of the source). Prove that for every prime $N \equiv 3 \pmod 4$ with $N > 3$ (equivalently, for every $N$ in the domain of the Chowla–Selberg formula above with odd class number $h$), the Chowla–Selberg factor $\lambda$ is a unit of the Hilbert class field $H_N$.

(b) Class-invariant conjecture (Conjecture 2 of the source). Prove that for every square-free $N \equiv 3 \pmod 8$ with $\gcd(N,3)=1$: the numbers $f = f(N)$ and $g = g(N)$ obtained from the mod-64 signature are algebraic integers, and $H_N$ is generated over $K$ by at least one of $f, g$; and that for $N > 1099$: $H_N$ is generated over $K$ by both $f$ and $g$, and their minimal polynomials over $\mathbb{Q}$ have degree $h(-N)$ (for $N \le 1099$ there are seven cases in which one of the two generates only a proper subfield, e.g. the minimal polynomial of $g(715)$ is $x^2 + x - 1$ while $h = 4$).

(c) Conceptual explanation of the mod-64 rule. Establish from CM theory — for instance through the explicit Shimura-reciprocity action of the ray class group of conductor dividing 48 on the eta quotients entering $r$ and on the nested radicals defining $s$ — why the admissible signature $[S_1,S_2,S_3]$, and the asymptotic prefactor $\alpha(N) = \sqrt{2}\cos(k\pi/16)$ in $g = \alpha(N)\,e^{\pi\sqrt{N}/48} + o(1)$, depend only on the residue of $N$ modulo 64, thereby replacing the empirical table by a theorem.

(d) Composite unit criterion. Prove that $\lambda^2$ is a unit of $H_N$ for every composite square-free $N \equiv 3 \pmod 4$ that is a product of distinct primes greater than 3, and find a criterion determining for which such composite discriminants $\lambda$ itself is a unit — the source's data being that $N = 19019 = 7 \cdot 11 \cdot 13 \cdot 19$ yields $\lambda$ as a unit, while $N = 1771 = 7 \cdot 11 \cdot 23$ requires $\lambda^2$.

An answer may resolve any part by proof or by explicit counterexample (a counterexample to a stated claim refutes that part); the problem as a whole is settled when each part is proved or refuted. For (c) a derivation from CM theory establishing the mod-64 dependence suffices; for (d) an explicit criterion deciding for which composite discriminants $\lambda$ is a unit suffices. The source's further remark that the computational advantages of $g$ persist uniformly at large $N$ is an empirical performance observation expected to follow from (b) and (c), and is not part of the completion boundary.

References

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