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Journal Articles Journal of Number Theory Year : 2020

Hilbert Modular Polynomials

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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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Dates and versions

hal-01990298 , version 1 (23-01-2019)

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