Addendum to "Measured foliations and Hilbert 12th problem"
Igor V. Nikolaev
TL;DR
This addendum provides numerical evidence supporting the link between explicit class field theory for real quadratic fields and the Fourier-analytic data of Hecke eigenforms, extending results from Acta Mathematica Vietnamica 48 (2023). Using computational databases and tools, it demonstrates that for primes $p$ with $p\equiv 3\pmod 4$ there exist least conductors $\mathfrak{f}_1,\mathfrak{f}_2$ such that $\mathscr{H}(\mathbf{Q}(\sqrt{p}))\bmod\mathfrak{f}_1\subseteq K_f$ for $f\in S_2(\Gamma(m^2p))$, aligning with the predicted class-field-theoretic structure via the isomorphism of class groups. A detailed $p=7$ example with $N=63=3^2\cdot 7$ yields a degree-4 coefficient field $K_f$ with minimal polynomial $x^4-8x^2+9$ and an explicit ring class field realization $\mathscr{H}(\mathbf{Q}(\sqrt{7}))\bmod 3 \cong \mathbf{Q}(\pm\sqrt{4\pm\sqrt{7}})$, illustrating the mechanism through measured foliations and real multiplication. Overall, the paper provides concrete numerical instances that support the proposed framework and guide further computational investigations in explicit real-quadratic Hilbert 12th problem research.
Abstract
We study numerical examples of the abelian extensions of the real quadratic number fields based on the results in Acta Mathematica Vietnamica 48 (2023), 271-281 (arXiv:0804.0057)