Abstract
The image of a complex conjugation by a two-dimensional representation of \(\mathrm{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\) in characteristic 2 can be trivial or a non-trivial element of order 2. Since such a representation comes from a modular form f, we attempt to study such an alternative purely in terms of f. The component group of the real points of the modular Jacobian J 1(N) plays a role in this question. We give an elementary description of that group. Our method to obtain such a description applies to determine the component group at infinity of the Jacobian of modular curves over \(\mathbb {R}\) attached to any subgroup of finite index of \(\mathrm{SL}_{2}(\mathbb {Z})\). We show that such a group is always “Eisenstein”. We derive a few consequences of this last fact for Galois representations and modular parametrisations.
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Merel, L. (2014). Formes modulaires modulo 2 et composantes réelles de jacobiennes modulaires. In: Böckle, G., Wiese, G. (eds) Computations with Modular Forms. Contributions in Mathematical and Computational Sciences, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-03847-6_8
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DOI: https://doi.org/10.1007/978-3-319-03847-6_8
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