Infinite Hilbert Class Field Towers from Galois Representations

TL;DR

This paper constructs infinite class field towers from Galois representations associated with modular forms and elliptic curves, providing explicit examples and conditional infinite families under certain conjectures.

Contribution

It explicitly identifies primes leading to infinite class field towers from modular form Galois representations and establishes infinite towers for elliptic curve division fields under a conjecture.

Findings

Explicit primes for infinite towers from modular forms

Conditional proof of infinite towers assuming Hardy-Littlewood conjecture

Infinite towers for elliptic curve division fields for density-one sets of integers

Abstract

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 $M_E$.