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Nonsolvable number fields ramified only at 3 and 5

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields Arithmetic algebraic geometry Computational number theory

Published online by Cambridge University Press:  15 February 2011

Lassina Dembélé ,
Matthew Greenberg  and
John Voight
Lassina Dembélé
Affiliation:
Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: l.dembele@warwick.ac.uk)
Matthew Greenberg
Affiliation:
University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4 (email: mgreenbe@math.ucalgary.ca)
John Voight
Affiliation:
Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT 05401, USA (email: jvoight@gmail.com)
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Abstract

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For p=3 and p=5, we exhibit a finite nonsolvable extension of ℚ which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

Keywords

Galois theoryHilbert modular formsShimura curvesGalois representationscomputational number theory

MSC classification

Secondary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11G18: Arithmetic aspects of modular and Shimura varieties 11F80: Galois representations 11R32: Galois theory 11Y40: Algebraic number theory computations 11R52: Quaternion and other division algebras

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 716 - 734
Copyright
Copyright © Foundation Compositio Mathematica 2011

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