Most authors who give examples of number fields having an infinite Hilbert class field tower (IHCFT) start with a fixed number field \(K\), and construct extensions of \(K\) in which a larger number of primes ramify. They essentially use results from genus theory and variants of the Golod-Shafarevich inequality to complete their proofs. In the present paper, the authors investigate Hilbert class field towers as fixed fields of representations of the absolute Galois group \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\), arising from modular forms or elliptic curves without complex multiplication, and give new examples of number fields having an IHCFT.
Let \(k\in \{12, 16, 18, 20, 22, 26\}\). There exists a unique normalized cuspidal Hecke eigenform \(\Delta_k\) on \(\text{SL}_2(\mathbb{Z})\) of weight \(k\). A theorem of Deligne-Serre provides for each \(\Delta_k\) and prime \(l\) a Galois representation \(\rho_{\Delta_{k,l}} : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \text{GL}_2(\mathbb{F}_l)\) unramified outside \(l\). Let \(K_{\Delta_{k,l}}\) be the fixed field by \(\text{Ker}(\rho_{\Delta_{k,l}})\). Let us fix \(k\); by a result of Serre–Swinnerton-Dyer, \(\text{Gal}(K_{\Delta_{k,l}}/\mathbb{Q}) \simeq \{g\in \text{GL}_2(\mathbb{F}_l)\mid \det(g)\in (\mathbb{F}_{l}^\ast)^{k-1}\}\), except for a finite number of explicit (and known) \(l\), called exceptional for \(\Delta_k\). The authors prove that if \(l\) is not exceptional for \(\Delta_k\) and \(\gcd(k-1, l-1)=1\) (hence \(\text{Gal}(K_{\Delta_{k,l}}/\mathbb{Q})\simeq \text{GL}_2(\mathbb{F}_l)\)), and \(\mathbb{Q}(\zeta_l)\) has an IHCFT, then \(K_{\Delta_{k,l}}\) has an IHCFT.
An example is \(K_{\Delta_{12,877}}\), thanks to the fact that it is well known that \(\mathbb{Q}(\zeta_{877})\) has an IHCFT. Assuming a well known conjecture of Hardy and Littlewood on the existence of infinitely primes of the form \(h(x)\), where \(x\in \mathbb{Z}\), and \(h=aX^2+bX+c \in \mathbb{Z}[X]\), with \(a+b\) and \(c\) not both even and \(b^2-4ac\) not a square, the authors prove the existence of infinitely \(l\) such that \(\mathbb{Q}(\zeta_l)\) has an IHCFT. Therefore there exist infinitely many primes \(l\) such that \(K_{\Delta_{k,l}}\) has an IHCFT.
Now let \(E\) be an elliptic curve without complex multiplication. Let \(\rho_n : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \text{GL}_2(\mathbb{Z}/n\mathbb{Z})\) be the Galois representation associated to the \(n\)-torsion points of \(E\). Let \(K_n\) be the fixed field by \(\text{Ker}(\rho_n)\). An exceptional prime of \(E\) is a prime \(l\) such that \(\rho_l\) is not surjective; it is known that the number of such \(l\) is finite. Let \(A_E\) be the product of all exceptional primes of \(E\). It is also known that for all \(n\) prime to \(30A_E\), \(\rho_n\) is surjective, and hence \(\text{Gal}(K_n/K)\simeq \text{GL}_2(\mathbb{Z}/n\mathbb{Z})\). Let \(S\) be the set of the integers prime to \(30A_E\). The authors prove that for all \(n\in S\) outside a subset of density \(0\), \(K_n\) has an IHCFT.