Infinite Hilbert Class Field Towers from Galois Representations

Details Infinite Hilbert Class Field Towers from Galois Representations Infinite Hilbert Class Field Towers from Galois Representations

2010-05-17

arxiv.org/abs/1005.3003v1

International Journal of Number Theory, Vol. 7, No. 2. March, 2011

Mathematics

Number Theory

Scientific paper

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 $k\in\{12,16,18,20,22,26\}$, we give explicit rational primes $\l$ such that the fixed field of the mod-$\l$ representation attached to the unique normalized cusp eigenforms of weight $k$ on $\Sl_2(\Z)$ 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 $E/\Q$, 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 $M_E$.

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