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Even Galois representations and the cohomology of \(\mathrm{GL}(2,\mathbb Z)\)

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Abstract

Let \(\rho \) be an even two-dimensional representation of the Galois group \({{\mathrm{Gal}}}(\overline{\mathbb Q}/\mathbb Q)\) which is induced from a character \(\chi \) of odd order of the absolute Galois group of a real quadratic field K. After imposing some additional conditions on \(\chi \), we attach \(\rho \) to a Hecke eigenclass in the cohomology of \(\mathrm{GL}(2,\mathbb Z)\) with coefficients in a certain infinite-dimensional vector space V over an arbitrary field of characteristic not equal to 2. The space V is defined purely algebraically starting from the field K.

Résumé

Soit \(\rho \) une représentation de degré 2 du groupe de Galois \({{\mathrm{Gal}}}(\overline{\mathbb Q}/\mathbb Q)\) et de déterminant pair. On suppose que \(\rho \) est induite par un caractère \(\chi \) d’ordre impair appartenant à un corps quadratique réel. Avec conditions additionnelles sur \(\chi \), on démontre que \(\rho \) est attachée a un vecteur propre des opérateurs de Hecke dans la cohomologie de \(\mathrm{GL}(2,\mathbb Z)\) avec coefficients dans un certain espace vectoriel V de dimension infinie sur un corps quelconque de caractéristique non égale a 2. Cet espace V est construit purement algébriquement à partir du corps K.

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Authors and Affiliations

  1. Boston College, Chestnut Hill, MA, 02467, USA

    Avner Ash

  2. Brigham Young University, Provo, UT, 84602, USA

    Darrin Doud

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  1. Avner Ash
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  2. Darrin Doud
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Ash, A., Doud, D. Even Galois representations and the cohomology of \(\mathrm{GL}(2,\mathbb Z)\). Ann. Math. Québec 43, 1–35 (2019). https://doi.org/10.1007/s40316-018-0104-7

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