### 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 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 SL_{2}(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}.

Original language | English (US) |
---|---|

Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | International Journal of Number Theory |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2011 |

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### Keywords

- Class field tower
- Galois representation
- modular form

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*International Journal of Number Theory*,

*7*(1), 1-8. https://doi.org/10.1142/S1793042111003879

**Infinite hilbert class field towers from galois representations.** / Joshi, Kirti N; McLeman, Cameron.

Research output: Contribution to journal › Article

*International Journal of Number Theory*, vol. 7, no. 1, pp. 1-8. https://doi.org/10.1142/S1793042111003879

}

TY - JOUR

T1 - Infinite hilbert class field towers from galois representations

AU - Joshi, Kirti N

AU - McLeman, Cameron

PY - 2011/2

Y1 - 2011/2

N2 - 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 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.

AB - 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 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.

KW - Class field tower

KW - Galois representation

KW - modular form

UR - http://www.scopus.com/inward/record.url?scp=79952289744&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952289744&partnerID=8YFLogxK

U2 - 10.1142/S1793042111003879

DO - 10.1142/S1793042111003879

M3 - Article

VL - 7

SP - 1

EP - 8

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 1

ER -