Infinite hilbert class field towers from galois representations

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each κ ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes ℓ such that the fixed field of the mod-ℓ representation attached to the unique normalized cusp eigenform of weight κ on SL₂(ℤ) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes ℓ for each κ in the list. Finally, given a non-CM curve E/ℚ, we show that there exists an integer M_E such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to M_E.