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Hensel-lifting torsion points on Jacobians and Galois representations


Author: Nicolas Mascot
Journal: Math. Comp. 89 (2020), 1417-1455
MSC (2010): Primary 11F80, 11Y40, 14Q05, 14H40, 14G10, 14G15, 14G20
DOI: https://doi.org/10.1090/mcom/3484
Published electronically: November 4, 2019
MathSciNet review: 4063323
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Abstract: Let $ \rho $ be a mod $ \ell $ Galois representation. We show how to compute $ \rho $ explicitly, given the characteristic polynomial of the image of the Frobenius at one prime $ p$ and a curve $ C$ whose Jacobian contains $ \rho $ in its $ \ell $-torsion. The main ingredient is a method to $ p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.


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Additional Information

Nicolas Mascot
Affiliation: Department of Mathematics, American University of Beirut, Beirut, Lebanon
Address at time of publication: School of Mathematics, Trinity College Dublin, Dublin, Ireland
Email: mascotn@tcd.ie

DOI: https://doi.org/10.1090/mcom/3484
Keywords: Galois representation, Jacobian, $p$-adic, algorithm
Received by editor(s): November 6, 2018
Received by editor(s) in revised form: March 29, 2019, and June 6, 2019
Published electronically: November 4, 2019
Article copyright: © Copyright 2019 American Mathematical Society