Infinite Hilbert Class Field Towers from Galois Representations

الأصل · EN

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 k∈{12,16,18,20,22,26}, we give explicit rational primes ł such that the fixed field of the mod-ł representation attached to the unique normalized cusp eigenforms of weight k on ₂() has an infinite class field tower. Under a conjecture of Hardy and Littlewood, we further prove that there exist infinitely many such primes for each k (in the above list). Second, given a non-CM curve E/, we show that there exists an integer Mₑ 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 Mₑ.