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We investigate class field towers of number fields obtained as fixed fieldsof modular representations of the absolute Galois group of the rationalnumbers. First, for each $k\in\{12,16,18,20,22,26\}$, we give explicit rationalprimes $\l$ such that the fixed field of the mod-$\l$ representation attachedto the unique normalized cusp eigenforms of weight $k$ on $\Sl_2(\Z)$ has aninfinite class field tower. Under a conjecture of Hardy and Littlewood, wefurther prove that there exist infinitely many such primes for each $k$ (in theabove list). Second, given a non-CM curve $E/\Q$, we show that there exists aninteger $M_E$ 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 ofintegers $n$ of density one among integers coprime to $M_E$.

INFINITE HILBERT CLASS FIELD TOWERS FROM GALOIS REPRESENTATIONS