Infinite hilbert class field towers from galois representations

Kirti Joshi, Cameron McLeman

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)1-8
Number of pages8
JournalInternational Journal of Number Theory
Volume7
Issue number1
DOIs
StatePublished - Feb 1 2011

Keywords

  • Class field tower
  • Galois representation
  • modular form

ASJC Scopus subject areas

  • Algebra and Number Theory

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