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Hilbert Modular Polynomials

Abstract : We present an algorithm to compute a higher dimensional analogue of modular polynomials. This higher dimensional analogue, the 'set of Hilbert modular polynomials', concerns cyclic isogenies of principally polarised abelian varieties with maximal real multiplication by a fixed totally real number field K0. We give a proof that this algorithm is correct, and provide practical improvements and an implementation for the 2-dimensional case with K0 = Q(√ 5). We also explain applications of this algorithm to point counting, walking on isogeny graphs, and computing class polynomials.
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Submitted on : Wednesday, January 23, 2019 - 6:45:52 AM
Last modification on : Tuesday, October 19, 2021 - 11:06:00 PM
Long-term archiving on: : Wednesday, April 24, 2019 - 12:59:59 PM

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Chloe Martindale. Hilbert Modular Polynomials. Journal of Number Theory, Elsevier, 2020, 213, pp.464-498. ⟨10.1016/j.jnt.2019.11.019⟩. ⟨hal-01990298⟩

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