JournalslemVol. 65, No. 3/4pp. 271–301

A note on Galois representations with big image

  • Nicholas M. Katz

    Princeton University, USA
A note on Galois representations with big image cover

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Abstract

Given an integer N3N \ge 3, we will first construct motivic representations (i.e., built out of pieces of the cohomology of projective smooth varieties, in fact curves)

ρ:Gal(Q/Q(ζN))GL(n,Q)\rho: Gal(\overline{\mathbb Q}/\mathbb Q(\zeta_N)) \rightarrow GL(n,\mathbb Q_\ell)

with open image, for any \ell which is 11 mod NN and for certain nn. We will do this in three different ways. The third of them has a descent to Q\mathbb Q when NN is 3 or 4. This provides us with motivic Galois representations of Gal(Q/Q)Gal(\overline{\mathbb Q}/\mathbb Q) with open image in GL(n,Q)GL(n,\mathbb Q_\ell) for any even n6n \ge 6 and any \ell which is 1\equiv 1 mod 3 or mod 4.

Cite this article

Nicholas M. Katz, A note on Galois representations with big image. Enseign. Math. 65 (2020), no. 3, pp. 271–301

DOI 10.4171/LEM/65-3/4-1