Research PaperNo Access

Images of Galois representations in mod $p$ Hecke algebras

Laia Amorós

Department of Computer Science, Aalto University, Maarintie 8, Espoo 02150, Finland

https://doi.org/10.1142/S1793042121500354Cited by:1

Abstract

Let $(𝕋_{f}, 𝔪_{f})$ denote the mod $p$ local Hecke algebra attached to a normalized Hecke eigenform $f$ , which is a commutative algebra over some finite field $𝔽_{q}$ of characteristic $p$ and with residue field $𝔽_{q}$ . By a result of Carayol we know that, if the residual Galois representation ${\bar{ρ}}_{f} : G_{ℚ} \to {GL}_{2} (𝔽_{q})$ is absolutely irreducible, then one can attach to this algebra a Galois representation $ρ_{f} : G_{ℚ} \to {GL}_{2} (𝕋_{f})$ that is a lift of ${\bar{ρ}}_{f}$ . We will show how one can determine the image of $ρ_{f}$ under the assumptions that (i) the image of the residual representation contains ${SL}_{2} (𝔽_{q})$ , (ii) $𝔪_{f}^{2} = 0$ and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain $p$ -elementary abelian extensions of big non-solvable number fields.

Keywords:

AMSC: 11F80, 11F33, 20E34

References

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Vol. 17, No. 05

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History

Received 31 July 2019

Revised 7 February 2020

Accepted 28 September 2020

Published: 16 November 2020

Keywords

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Images of Galois representations in mod p Hecke algebras

Abstract

References

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Images of Galois representations in mod $p$ Hecke algebras