Extended Hecke class-field inclusions and missing ring class field data in the measured-foliation approach to Hilbert's 12th problem

In a 2025 addendum to his measured-foliation approach to Hilbert's 12th problem for real quadratic fields, Nikolaev proves that for every prime $p\equiv 3\pmod 4$ with $p\ge 7$ the ring class fields of $\mathbb{Q}(\sqrt{p})$ of a suitable conductor embed into the Fourier coefficient fields of weight-2 Hecke eigenforms of level $m^2p$, and he tabulates supporting data for primes $7\le p\le 163$. The addendum explicitly leaves two linked issues unresolved: a rigorous general proof of the Hecke-type inclusions beyond the classically established case — namely their extension to Pythagorean primes $p\equiv 1\pmod 4$ and to composite levels, which is stated to be 'unknown to the author' (Remark 2) — and the missing computational data for the primes $p=71,79,131,151$, where least conductors, exact levels, and defining polynomials of the relevant ring class fields exceed the capacity of the consulted databases (Remark 7, Table 6). This record presents these two open issues as one problem: to put the foliation-based realization of ring class fields of real quadratic fields on a rigorous footing for the extended levels and to complete the numerical evidence for the four listed primes.

Background

Hilbert's twelfth problem asks for an explicit construction of the maximal abelian extension of a number field, in the spirit of the Kronecker–Weber theorem (abelian extensions of $\mathbb{Q}$ are generated by roots of unity) and the theory of complex multiplication (abelian extensions of an imaginary quadratic field are generated by special values of modular functions, such as the $j$-invariant, a modular function that classifies elliptic curves; for example, the Hilbert class field — the maximal abelian extension unramified at all primes — of an imaginary quadratic field is generated by a single value of $j$). For real quadratic fields no comparable explicit theory is known. Hecke, in his 1910 Göttingen thesis written under Hilbert and in a 1928 paper, used automorphic functions to construct class fields of real quadratic fields; the addendum quotes the following consequence (Hecke 1928, p. 709): for every prime $p\equiv 3\pmod 4$ with $p\ge 7$ there exists a Hecke eigenform $f$ in the space $S_2(\Gamma(p))$ of weight-2 cusp forms for the principal congruence subgroup $\Gamma(p)$ of $\mathrm{SL}_2(\mathbb{Z})$ (the kernel of reduction modulo $p$) — cusp forms being holomorphic functions on the upper half-plane transforming by $f((az+b)/(cz+d))=(cz+d)^2f(z)$ and vanishing at the cusps of the modular curve $X(p)$ — such that the field generated by its Fourier coefficients, $K_f=\mathbb{Q}(\{c_n(f)\})$, contains the Hilbert class field $H(\mathbb{Q}(\sqrt{-p}))$ of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$. Here a Hecke eigenform is a cusp form that is a simultaneous eigenvector for the Hecke operators $T_n$, the natural integral correspondences acting on modular forms; more generally, the ring class field of a quadratic field modulo an integer conductor $c\ge 1$ is the abelian extension whose Galois group is canonically isomorphic to the ring class group $\mathrm{Cl}(k\bmod c)$ of the order of conductor $c$ (the case $c=1$ recovers the Hilbert class field). Manin proposed that the missing explicit class field theory for real quadratic fields should involve noncommutative geometry ('real multiplication'); Nikolaev's program realizes this idea through measured foliations on modular curves — decompositions of the surface into leaves carrying a transverse measure, obtained here as the real part of the holomorphic differential $f(z)\,dz$. Theorem 3 of the 2023 base paper states that for every level $N>1$ some real quadratic field $\mathbb{Q}(\sqrt{D})$ and conductor $c\ge 1$ satisfy $H(\mathbb{Q}(\sqrt{D}))\bmod c\subseteq K_f$, and the 2016 predecessor paper relates the level to the discriminant. The addendum's Theorem 4 refines this for primes $p\equiv 3\pmod 4$, $p\ge 7$: using the least non-trivial conductors $c_1,c_2\ge 1$ (non-trivial meaning $c_1,c_2>1$, so that the degenerate pair $(1,1)$ is excluded, and least taken lexicographically by $(c_1,c_2)$) for which the ring class groups are isomorphic, $\mathrm{Cl}(\mathbb{Q}(\sqrt{p})\bmod c_1)\cong\mathrm{Cl}(\mathbb{Q}(\sqrt{-p})\bmod c_2)$, it asserts that $H(\mathbb{Q}(\sqrt{p}))\bmod c_1$ embeds into $K_f$ for a Hecke eigenform of level $N=m^2p$. Example 5 carries this out for $p=7$: $c_1=3$, $c_2=4$, common group $\mathbb{Z}/2\mathbb{Z}$, level $63=3^2\cdot 7$, and the passage from the coefficient field (minimal polynomial $x^4+8x^2+9$, with purely imaginary roots $\pm i\sqrt{4\pm\sqrt{7}}$) to the measured foliation (extracting the imaginary parts, i.e. the real quantities $\sqrt{4\pm\sqrt{7}}$) yields $H(\mathbb{Q}(\sqrt{7})\bmod 3)\cong\mathbb{Q}\left(\pm\sqrt{4\pm\sqrt{7}}\right)$. Table 6 records analogous data for $7\le p\le 163$, computed with the databases lmfdb.org (the L-functions and Modular Forms Database), numbertheory.org, the computer algebra system PARI/GP, and the online Magma calculator. Two issues are explicitly left open. Remark 2 states that the Hecke inclusion above extends to the Pythagorean primes $p\equiv 1\pmod 4$ (citing the numerical Tables Ia and Ib of Shimura's 1972 Annals paper on class fields over real quadratic fields and Hecke operators) and, in a modified form, to composite levels $N$ (citing Hecke's 1928 paper), and that a rigorous proof of these extended inclusions is 'unknown to the author'. Section 4 of the 2023 base paper records the form in which Shimura's Pythagorean-prime result is proved: for a prime level $N=p\equiv 1\pmod 4$ one considers Hecke eigenforms $f(z,\psi)\in S_2(\Gamma_1(p),\psi)$ carrying the quadratic character $\psi(a)=(a/p)$, whose coefficient field $K_{f,\psi}$ is a CM field (Shimura 1972, Proposition 1.3) whose maximal real subfield $F$ is interpreted as a ray class field of the real quadratic field $k=\mathbb{Q}(\sqrt{p})$ attached to an integral ideal $\mathfrak{c}$, with conductor $N(\mathfrak{c})$; Shimura's numerical examples (the Tables Ia and Ib cited by Remark 2) cover levels $29\le N\le 233$. The Pythagorean-prime extension evidenced by Remark 2 therefore lives in a modified normalization — level $p$ for $\Gamma_1(p)$ with character, and ray class fields of $\mathbb{Q}(\sqrt{p})$ — rather than in the $\Gamma(p)$, Hilbert-class-field-of-$\mathbb{Q}(\sqrt{-p})$ form of the 1928 inclusion. Remark 7 states that the Table 6 entries for $p=71,79,131,151$ exceed the current capacity of lmfdb.org and/or Magma: for $p=71,79$ even the least conductors are known only through the bounds $c_1>50$, $c_2>10$, while for $p=131,151$ the conductors and ring class groups are tabulated but the exact level $m$ and the full defining polynomials of the ring class fields are not.

Problem Statement

The problem consists of two linked components sharing one objective: to put the measured-foliation realization of ring class fields of real quadratic fields — the addendum's Theorem 4 — on a complete and rigorous footing for the levels and primes it actually needs. Component (A) — rigorous extended Hecke inclusions: prove or refute, with complete proofs, the extended inclusions asserted in Remark 2 of the addendum — namely that the Hecke inclusion of the 1928 theorem (for every prime $p\equiv 3\pmod 4$, $p\ge 7$, there is a Hecke eigenform $f\in S_2(\Gamma(p))$ whose coefficient field $K_f$ contains the Hilbert class field of $\mathbb{Q}(\sqrt{-p})$) extends (a) to the Pythagorean primes $p\equiv 1\pmod 4$ and (b) to composite levels $N$, in particular the levels $N=m^2p$ used in Theorem 4, for which the addendum's proof asserts without justification that 'inclusion (1) is also true for the levels $\{m^2p\mid m\ge 1\}$' and for which Remark 2 records that 'a rigorous proof of these inclusions is unknown to the author'. Because the addendum alludes to the exact statements of these extensions only '(in a modified form)' and refers to the cited literature — Shimura's 1972 paper (whose Tables Ia and Ib supply the numerical evidence for the Pythagorean-prime case) and Hecke's 1928 paper — the exact inclusion to be proved or refuted is the one asserted in those references, specifying the level group, the quadratic field involved (real or imaginary), whether Hilbert, ray or ring class fields and at which conductors, and the containment asserted in the coefficient field $K_f$, in the form the references assert, and not a weaker variant that the cited references do not assert — such as a naive $\Gamma(p)$ extension with complex multiplication by $\mathbb{Q}(\sqrt{-p})$ for $p\equiv 1\pmod 4$. Component (B) — completion of Table 6: for $p=71$ and $p=79$, determine the least non-trivial conductors $c_1,c_2\ge 1$ ($c_1,c_2>1$, least lexicographically by $(c_1,c_2)$) with $\mathrm{Cl}(\mathbb{Q}(\sqrt{p})\bmod c_1)\cong\mathrm{Cl}(\mathbb{Q}(\sqrt{-p})\bmod c_2)$ and the common ring class group in invariant-factor form (currently known only through the bounds $c_1>50$, $c_2>10$); and for each of $p=71,79,131,151$, determine the least $m>10$ for which a Hecke eigenform of level $N=m^2p$ has coefficient field containing the ring class field $H(\mathbb{Q}(\sqrt{p}))\bmod c_1$ (the exact level recorded as missing in Table 6 is this least admissible $m$, currently known only through the bound $m>10$), and give explicitly the full defining polynomial of that ring class field (for $p=131$ and $p=151$ the conductors and groups are already tabulated, and the polynomials are currently truncated as $x^{80}+\cdots$ and $x^{56}+\cdots$). No restrictions on method (proof technique, software, or database) are imposed.

References

1. Igor V. Nikolaev, Addendum to "Measured foliations and Hilbert 12th problem", arXiv preprint arXiv:2505.22272 (2025), doi:10.48550/arXiv.2505.22272, https://arxiv.org/abs/2505.22272
2. Igor Nikolaev, Measured foliations and Hilbert 12th problem, Acta Mathematica Vietnamica 48 (2023) 271–281, doi:10.1007/s40306-022-00479-w, arXiv:0804.0057, https://arxiv.org/abs/0804.0057 (Zbl 1540.11085)
3. Igor V. Nikolaev, Real multiplication revisited, Journal of Generalized Lie Theory and Applications 10 (2016) S2-007, doi:10.4172/1736-4337.1000S2-007, arXiv:1404.4999, https://arxiv.org/abs/1404.4999
4. Erich Hecke, Bestimmung der Perioden gewisser Integrale durch die Theorie der Klassenkörper, Mathematische Zeitschrift 28 (1928) 708–727, doi:10.1007/BF01181192, https://doi.org/10.1007/BF01181192 (JFM 54.0405.01)
5. Erich Hecke, Über die Konstruktion der Klassenkörper reeller quadratischer Körper mit Hülfe von automorphen Funktionen, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1910) 619–623, JFM 41.0244.04, https://zbmath.org/?q=an:JFM+41.0244.04
6. Goro Shimura, Class fields over real quadratic fields and Hecke operators, Annals of Mathematics (2) 95 (1972) 130–190, doi:10.2307/1970859, https://doi.org/10.2307/1970859 (Zbl 0255.10032)
7. Koji Doi and Masatoshi Yamauchi, On the Hecke operators for Γ₀(N) and class fields over quadratic number fields, Journal of the Mathematical Society of Japan 25 (1973) 629–643, doi:10.2969/jmsj/02540629, https://doi.org/10.2969/jmsj/02540629 (Zbl 0259.10024)
8. Benedict H. Gross, On Hecke's decomposition of the regular differentials on the modular curve of prime level, Research in the Mathematical Sciences 5 (2018) Paper No. 1, doi:10.1007/s40687-018-0121-9, https://doi.org/10.1007/s40687-018-0121-9
9. Yu. I. Manin, Real Multiplication and Noncommutative Geometry (ein Alterstraum), in: The Legacy of Niels Henrik Abel, Springer, Berlin (2004) 685–727, doi:10.1007/978-3-642-18908-1_23, https://doi.org/10.1007/978-3-642-18908-1_23