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7 - Galois realizations of families of Projective Linear Groups via cusp forms

Published online by Cambridge University Press:  08 October 2009

Bas Edixhoven
Affiliation:
Universiteit Leiden
Gerard van der Geer
Affiliation:
Universiteit van Amsterdam
Ben Moonen
Affiliation:
Universiteit van Amsterdam
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Summary

Introduction

Let Sk be the space of cusp forms of weight k for SL2(ℤ) and write S2(N) for the space of cusp forms of weight 2 for Γ0(N).

We are going to consider the Galois representations attached to eigenforms in these spaces, whose images have been determined by Ribet and Momose (see [Ri 75] for Sk and [Mo 81], [Ri 85] for S2(N)).

Our purpose is to use these representations to realize as Galois groups over ℚ some linear groups of the following form: PSL2(pr) if r is even and PGL2(pr) if r is odd. In order to ease the notation, we will call both these families of linear groups PXL2(pr), so that PXL stands for PSL if r is even and PGL if r is odd.

Extending the results in [Re-Vi], where it is shown that for r ≤ 10 these groups are Galois groups over ℚ for infinitely many primes p, we will cover the cases r = 11, 13, 17 and 19, using the representations attached to eigenforms in Sk and again the cases 11 and 17 using the ones coming from S2(N).

We will give the explicit criterion for the case r = 3: for every prime p > 3 such that p ≡ 2, 3, 4, 5 mod 7 the group PGL2(p3) is a Galois group over ℚ.

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Publisher: Cambridge University Press
Print publication year: 2008

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